arXiv:0704.0061v2  [math.FA]  3 May 2007
INTERSECTION BODIES AND GENERALIZED
COSINE TRANSFORMS
BORIS RUBIN
Abstract. Intersection bodies represent a remarkable class of
geometric objects associated with sections of star bodies and in-
voking Radon transforms, generalized cosine transforms, and the
relevant Fourier analysis. The main focus of this article is interrela-
tion between generalized cosine transforms of different kinds in the
context of their application to investigation of a certain family of
intersection bodies, which we call λ-intersection bodies. The latter
include k-intersection bodies (in the sense of A. Koldobsky) and
unit balls of finite-dimensional subspaces of Lp-spaces. In particu-
lar, we show that restrictions onto lower dimensional subspaces of
the spherical Radon transforms and the generalized cosine trans-
forms preserve their integral-geometric structure. We apply this
result to the study of sections of λ-intersection bodies. New char-
acterizations of this class of bodies are obtained and examples are
given. We also review some known facts and give them new proofs.
Contents
1. Introduction.
2. Preliminaries.
3. Analytic families of the generalized cosine transforms.
4. Positive definite homogeneous distributions.
5. λ-intersection bodies.
6. Examples of λ-intersection bodies.
7. (q, ℓ)-balls.
8. The generalized cosine transforms and comparison of volumes.
9. Appendix.
2000 Mathematics Subject Classification. Primary 44A12; Secondary 52A38.
Key words and phrases. Spherical Radon transforms, cosine transforms, inter-
section bodies.
The research was supported in part by the NSF grant DMS-0556157 and the
Louisiana EPSCoR program, sponsored by NSF and the Board of Regents Support
Fund.
http://arxiv.org/abs/0704.0061v2
2 BORIS RUBIN
1. Introduction
This is an updated and extended version of our previous preprint
[R5].
Intersection bodies interact with Radon transforms and encompass
diverse classes of geometric objects associated to sections of star bodies.
The concept of intersection body was introduced in the remarkable
paper by Lutwak [Lu] and led to a breakthrough in the solution of the
long-standing Busemann-Petty problem; see [G], [K4], [Lu], [Z2] for
references and historical notes.
We remind some known facts that will be needed in the following.
An origin-symmetric (o.s.) star body in Rn, n ≥ 2, is a compact set
K with non-empty interior such that tK ⊂ K ∀t ∈ [0, 1], K = −K,
and the radial function ρK(θ) = sup{λ ≥ 0 : λθ ∈ K} is continuous
on the unit sphere Sn−1. In the following, Kn denotes the set of all
o.s. star bodies in Rn, Gn,i is the Grassmann manifold of i-dimensional
linear subspaces of Rn, and voli(·) denotes the i-dimensional volume
function. The Minkowski functional of a body K ∈ Kn is defined by
||x||K = min{a ≥ 0 : x ∈ aK}, so that ||θ||K = ρ−1K (θ), θ ∈ Sn−1.
Definition 1.1. [Lu] A body K ∈ Kn is an intersection body of a body
L ∈ Kn if ρK(θ) = voln−1(L ∩ θ⊥) for every θ ∈ Sn−1, where θ⊥ is the
central hyperplane orthogonal to θ.
By taking into account that voln−1(L ∩ θ⊥) in Definition 1.1 is a
constant multiple of the Minkowski-Funk transform
(Mf)(θ) =
Sn−1∩θ⊥
f(u) dθu, f(u) = ρ
L (u),
Goodey, Lutwak and Weil [GLW] generalized Definition 1.1 as follows.
Definition 1.2. A body K ∈ Kn is an intersection body if ρK = Mµ
for some even non-negative finite Borel measure µ on Sn−1.
A sequence of bodies Kj ∈ Kn is said to be convergent to K ∈ Kn
in the radial metric if lim
||ρKj − ρK ||C(Sn−1) = 0.
Proposition 1.3. The class of intersection bodies is the closure of the
class of intersection bodies of star bodies in the radial metric.
Proposition 1.4. If K is an intersection body in Rn, n > 2, then for
every i = 2, 3, . . . , n − 1 and every η ∈ Gn,i, K ∩ η is an intersection
body in η.
Regarding these two important propositions see [FGW], [GW] and
a nice historical survey in [G].
INTERSECTION BODIES 3
Different generalizations of the concept of intersection body associ-
ated to lower dimensional sections were suggested in the literature; see,
e.g., [K4], [RZ], [Z1]. The following one, which plays an important role
in the study of the lower dimensional Busemann-Petty problem, is due
to Zhang [Z1].
Definition 1.5. We say, that a body K ∈ Kn belongs to Zhang’s class
Zni if there is a non-negative finite Borel measure m on the Grassmann
manifold Gn,i such that ρ
K = R
im, where R
i is the dual spherical
Radon transform; see (2.2), (2.5).
Another generalization was suggested by Koldobsky [K2] and de-
scribed in detail in [K4]. This class of bodies will be our main concern.
Definition 1.6. [K4, p. 71] A body K ∈ Kn is a k-intersection body
of a body L ∈ Kn (we write K = IBk(L)) if
(1.1) volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) ∀ξ ∈ Gn,k.
We denote by IBk,n the set of all bodies K ∈ Kn satisfying (1.1) for
some L ∈ Kn.
When k = 1, this definition coincides with Definition 1.1 up to a
constant multiple. An analog of Definition 1.2 was given in the Fourier
analytic terms as follows.
Definition 1.7. [K4, Definition 4.7] A body K ∈ Kn is a k-intersection
body if there is a non-negative finite Borel measure µ on Sn−1, so that
for every Schwartz function φ,
||x||−kK φ(x) dx =
tk−1φ̂(tθ) dt
dµ(θ),
where φ̂ denotes the Fourier transform of φ.
The set of all k-intersection bodies in Rn will be denoted by Ink .
Keeping in mind Proposition 1.3 for k = 1, one can alternatively
define the class Ink as a closure of IBk,n in the radial metric; cf. [Mi1,
p. 532]. However, to apply results from [K4] to such class, equivalence
of this definition to Definition 1.7 must be proved. We will do this in
the more general situation in Section 5.2.
From Definitions 1.6 and 1.7 it is not clear, for which bodies L ∈ Kn
the relevant k-intersection body K = IBk(L) does exist. It is also not
obvious which bodies actually constitute the class Ink . The following
important characterization is due to Koldobsky.
4 BORIS RUBIN
Theorem 1.8. [K4, Theorem 4.8] A body K ∈ Kn is a k-intersection
body if and only if || · ||−kK represents a positive definite tempered dis-
tribution on Rn, that is, the Fourier transform (|| · ||−kK )∧ is a positive
tempered distribution on Rn.
The concept of k-intersection body is related to another important
development. For K ∈ Kn, the quasi-normed space (Rn, || · ||K) is said
to be isometrically embedded in Lp, p > 0, if there is a linear operator
T : Rn → Lp([0, 1]) so that ||x||K = ||Tx||Lp([0,1]).
Theorem 1.9. [K4, Theorem 6.10] The space (Rn, || · ||K) embeds iso-
metrically in Lp, p > 0, p 6= 2, 4, . . . , if and only if Γ(−p/2)(|| · ||pK)∧
is a positive distribution on Rn \ {0}.
Following Theorems 1.9 and 1.8, one can formally say that K ∈ Ink if
and only if (Rn, || · ||K) embeds isometrically in L−k. This observation,
combined with Definition 1.7, was used by A. Koldobsky to define the
concept of “isometric embedding in Lp” for negative p.
Definition 1.10. [K4, Definition 6.14] Let 0 < p < n, K ∈ Kn. The
space (Rn, || · ||K) is said to be isometrically embedded in L−p if there
is a non-negative finite Borel measure µ on Sn−1, so that for every
Schwartz function φ,
||x||−pK φ(x) dx =
tp−1φ̂(tθ) dt
dµ(θ),
where φ̂ denotes the Fourier transform of φ.
Origin-symmetric bodiesK in this definition can be regarded as “unit
balls of n-dimensional subspaces of L−p”. Comparing Definitions 1.10
and 1.7, one might call these bodies “p-intersection bodies”. Since the
meaning of the space L−p itself is not specified in Definition 1.10 and
since our paper is mostly focused on geometric properties of bodies
(rather than embeddings in Lp), in the following we prefer to adopt
another name “λ-intersection body”, where λ is a real number, that
will be specified in due course. We denote the set of all λ-intersection
bodies in Rn by Inλ .
Contents of the paper. We will focus on intimate connection
between intersection bodies, spherical Radon transforms, and general-
ized cosine transforms; see definitions in Section 2.2. This approach
is motivated by the fact that the volume of a central cross section of
a star body is expressed through the spherical Radon transform, and
the latter is a member of the analytic family of the generalized cosine
transforms. These transforms were introduced by Semyanistyi [Se] and
INTERSECTION BODIES 5
arise (up to naming and normalization) in different contexts of analysis
and geometry; see, e.g., [K4], [R1]-[RZ], [Sa2], [Sa3], [Str1], [Str2].
Sections 2-4 provide analytic background for geometric considera-
tions in Sections 5-7. In Section 2 we establish our notation and define
the generalized cosine transforms on the sphere and the relevant dual
transforms on Grassmann manifolds. In Section 3 we present basic
properties of these transforms, establish new relations between spheri-
cal Radon transforms and the generalized cosine transforms, and prove
“restriction theorems”, which are akin to trace theorems in Sobolev
spaces. Section 4 deals with positive definite homogeneous distribu-
tions, that can be characterized in terms of the generalized cosine
transforms. This section serves as a preparation for the forthcoming
definition of the concept of λ-intersection body. We investigate which
λ’s are appropriate and why. In Section 5 we switch to geometry and
define the class Inλ of λ-intersection bodies. The case 0 < λ < n cor-
responds to the “unit balls of L−p-spaces” in the spirit of Definition
1.10. The reader will find in this section new proofs of some known
facts. We introduce the notion of λ-intersection body of a star body
in Rn, which extends Definition 1.6 to all λ < n, λ 6= 0. The class
of all such bodies will be denoted by IBnλ . We will prove that for
all λ < n, λ 6= 0,−2,−4, . . . , the class Inλ is the closure of IBnλ in
the radial metric. The case λ = 1 gives Proposition 1.3. It will be
proved that all m-dimensional central sections of λ-intersection bod-
ies are λ-intersection bodies in the corresponding m-planes provided
λ < m, λ 6= 0.
The natural question arises: How to construct λ-intersection bodies?
In Section 6 we give a series of examples; some of them are known and
some are new. They can be obtained by utilizing auxiliary statements
from Section 3. In particular, the famous embedding of Zhang’s class
Znn−k into Ink , which was first established in [K3] and studied in [Mi1],
[Mi2], will be generalized to the case, when k is replaced by any λ ∈
(0, n). Section 7 is devoted to the so called (q, ℓ)-balls, defined by
Bnq,ℓ = {x = (x′, x′′) : |x′|q + |x′′|q ≤ 1; x′ ∈ Rn−ℓ, x′′ ∈ Rℓ}, q > 0.
We show that if 0 < q ≤ 2, then Bnq,ℓ ∈ Inλ for all λ ∈ (0, n). If
q > 2 and n − 3 ≤ λ < n, we still have Bnq,ℓ ∈ Inλ . If q > 2 and
0 < λ < λ0 = max(n − ℓ, ℓ) − 2, then Bnq,ℓ 6∈ Inλ . The case, when
q > 2, ℓ > 1, and λ0 ≤ λ < n− 3 represents an open problem.
In Section 8 we remind the generalized Busemann-Petty problem
(GBP) for i-dimensional central sections of o.s. convex bodies in Rn.
This challenging problem is still open for i = 2 and i = 3 (n ≥ 5).
It actually inspires the whole investigation. Using properties of the
6 BORIS RUBIN
generalized cosine transforms, we give a short direct proof of the fact
that an affirmative answer to GBP implies that every smooth o.s. con-
vex body in Rn with positive curvature is an (n− i)-intersection body.
This fact was discovered by A. Koldobsky. The original proof in [K3]
is based on the embedding Inn−i ⊂ Zni and Zhang’s result [Z1, Theorem
6]. The latter heavily relies on the Hahn-Banach separation theorem.
Our proof is more constructive and almost self-contained. We conclude
the paper by Appendix, which is added for convenience of the reader.
The list of references at the end of the paper is far from being com-
plete. Further references can be found in cited books and papers.
Acknowledgement. I am grateful to Professor Alexander Koldob-
sky, who shared with me his knowledge of the subject. Special thanks
go to Professors Erwin Lutwak, Deane Yang, and Gaoyong Zhang for
useful discussions.
2. Preliminaries
2.1. Notation. In the following, N = {1, 2, . . . } is the set of all nat-
ural numbers, Sn−1 is the unit sphere in Rn with the area σn−1 =
2πn/2/Γ(n/2); Ce(S
n−1) is the space of even continuous functions on
Sn−1; SO(n) is the special orthogonal group of Rn; for θ ∈ Sn−1 and
γ ∈ SO(n), dθ and dγ denote the relevant invariant probability mea-
sures; D(Sn−1) is the space of C∞-functions on Sn−1 equipped with
the standard topology, and D′(Sn−1) stands for the corresponding dual
space of distributions. The subspaces of even test functions (distribu-
tions) are denoted by De(Sn−1) ( D′e(Sn−1)); Gn,i denotes the Grass-
mann manifold of i-dimensional subspaces ξ of Rn with the SO(n)-
invariant probability measure dξ; D(Gn,i) is the space of infinitely dif-
ferentiable functions on Gn,i.
We write M(Sn−1) and M(Gn,i) for the spaces of finite Borel mea-
sures on Sn−1 and Gn,i; M+(Sn−1) and M+(Gn,i) are the relevant
spaces of non-negative measures; Me+(Sn−1) denotes the space of even
measures µ ∈ M+(Sn−1). Given a function ϕ on Gn,i, we denote
ϕ⊥(η) = ϕ(η⊥), η ∈ Gn,n−i. Similarly, given a measure µ ∈ M(Gn,n−i),
the corresponding “orthogonal measure” µ⊥ in M(Gn,i) is defined by
(µ⊥, ϕ) = (µ, ϕ⊥), ϕ ∈ C(Gn,i).
Let {Yj,k} be an orthonormal basis of spherical harmonics on Sn−1.
Here j = 0, 1, 2, . . . , and k = 1, 2, . . . , dn(j), where dn(j) is the di-
mension of the subspace of spherical harmonics of degree j. Each
function ω ∈ D(Sn−1) admits a decomposition ω =
j,k ωj,kYj,k with
the Fourier-Laplace coefficients ωj,k =
ω(θ)Yj,k(θ)dθ, which decay
rapidly as j → ∞. Each distribution f ∈ D′(Sn−1) can be defined by
INTERSECTION BODIES 7
(f, ω) =
j,k fj,kωj,k where fj,k = (f, Yj,k) grow not faster than j
m for
some integer m. We will need the Poisson integral, which is defined for
f ∈ L1(Sn−1) by
(2.1) (Πtf)(θ) = (1− t2)
f(u)|θ − tu|−ndu, 0 < t < 1,
and has the Fourier-Laplace decomposition Πtf =
j,k t
jfj,kYj,k [SW].
For f ∈ D′(Sn−1), this decomposition serves as a definition of Πtf . For
harmonic analysis on the unit sphere, the reader is referred to [Le],
[Mü], [Ne], [SW], and a survey article [Sa3].
2.2. Basic integral transforms. For integrable functions f on Sn−1
and ϕ onGn,i, 1 ≤ i ≤ n−1, the spherical Radon transform (Rif)(ξ), ξ ∈
Gn,i, and its dual (R
iϕ)(θ), θ ∈ Sn−1, are defined by
(2.2) (Rif)(ξ) =
θ∈Sn−1∩ξ
f(θ) dξθ, (R
iϕ)(θ) =
ϕ(ξ) dθξ,
where dξθ and dθξ denote the probability measures on the manifolds
Sn−1 ∩ ξ and {ξ ∈ Gn,i : ξ ∋ θ}, respectively. The precise meaning of
the second integral is
(2.3) (R∗iϕ)(θ) =
SO(n−1)
ϕ(rθγp0) dγ, θ ∈ Sn−1,
where p0 is an arbitrarily fixed coordinate i-plane containing the north
pole en and rθ ∈ SO(n) is a rotation satisfying rθen = θ.
Operators Ri and R
i extend to finite Borel measures in a canonical
way, using the duality
(2.4)
(Rif)(ξ)ϕ(ξ)dξ =
f(θ)(R∗iϕ)(θ)dθ.
Specifically, for µ ∈ M(Sn−1) and m ∈ M(Gn,i), we define Riµ ∈
M(Gn,i) and R∗im ∈ M(Sn−1) by
(2.5) (Riµ, ϕ)=
(R∗iϕ)(θ)dµ(θ), (R
im, f)=
(Rif)(ξ)dm(ξ),
where ϕ ∈ C(Gn,i), f ∈ C(Sn−1).
The generalized cosine transforms are defined by
(2.6) (Rαi f)(ξ) = γn,i(α)
|Prξ⊥θ|α+i−n f(θ) dθ,
(2.7) (
αϕ)(θ) = γn,i(α)
|Prξ⊥θ|α+i−n ϕ(ξ) dξ,
8 BORIS RUBIN
γn,i(α) =
σn−1 Γ((n− α− i)/2)
2π(n−1)/2 Γ(α/2)
, Re α > 0, α+i−n 6= 0, 2, 4, . . . .
Here Prξ⊥θ stands for the orthogonal projection of θ onto ξ
⊥, the or-
thogonal complement of ξ ∈ Gn,i. If f and ϕ are smooth enough, then
integrals (2.2) can be regarded (up to a constant multiple) as members
of the relevant analytic families (2.6) and (2.7); cf. Lemma 3.1. The
particular case i = n − 1 in (2.2) corresponds to the Minkowski-Funk
transform
(2.8) (Mf)(u) =
{θ : θ·u=0}
f(θ) duθ = (Rn−1f)(u
⊥), u ∈ Sn−1,
which integrates a function f over great circles of codimension 1. This
transform is a member of the analytic family
(2.9) (Mαf)(u) = (Rαn−1f)(u
⊥) = γn(α)
f(θ)|θ · u|α−1 dθ,
(2.10) γn(α)=
σn−1 Γ
(1−α)/2
2π(n−1)/2Γ(α/2)
, Re α>0, α 6=1, 3, 5, . . . .
The values α = 1, 3, 5, . . . are poles of the Gamma function Γ((1−α)/2).
In some occasions we include these values into consideration and set
(2.11) (M̃αf)(u) =
f(θ)|θ · u|α−1 dθ.
Historical notes. Regarding spherical Radon transforms (2.2) and
the Minkowski-Funk transform (2.8), see [GGG], [He], [R2], [R3]. The
first detailed investigation of the analytic family {Mα} is due to Se-
myanistyi [Se], who showed that these operators naturally arise in the
Fourier analysis of homogeneous functions. The case α = 2 in (2.11)
was known before, thanks to W. Blaschke, A.D. Alexandrov, and P.
Lévy. Integrals (2.9) (sometimes with different normalization) arise
in diverse areas of analysis and geometry; see [K4], [R1] - [R3], [Sa3],
[Str1], and references therein. In convex geometry and Banach space
theory, operators (2.11) with α − 1 replaced by p are known as the p-
cosine transforms. More general analytic families (2.6) and (2.7) were
introduced in [R2].
INTERSECTION BODIES 9
3. Analytic Families of the Generalized Cosine
Transforms
3.1. Basic properties. Below we review basic properties of integrals
(2.6), (2.7), (2.9); see [R2], [R3] for more details. For integrable func-
tions f and ϕ and Reα > 0, integrals (2.6), (2.7) and (2.9) are ab-
solutely convergent. When f and ϕ are infinitely differentiable, these
integrals extend meromorphically to all α ∈ C.
Lemma 3.1. If f and ϕ are continuous functions, then
Rαi f = R
i f = ciRif, ci =
2π(i−1)/2
;(3.1)
0ϕ = ciR
iϕ,(3.2)
Mαf = M0f = cn−1Mf, cn−1 =
2π(n−2)/2
.(3.3)
Hence, the Radon transform, its dual, and the Minkowski-Funk trans-
form can be regarded (up to a constant multiple) as members of the
corresponding analytic families {Rαi }, {
α}, {Mα}.
Proof. Formulas (3.2) and (3.3) follow from (3.1). To prove (3.1), we
write (2.6) in bi-spherical coordinates θ = u sin ψ + vcosψ, where
u ∈ Sn−1 ∩ ξ ∼ Si−1, v ∈ Sn−1 ∩ ξ⊥ ∼ Sn−i−1, 0 ≤ ψ ≤ π/2.
dθ = c sini−1 ψ cosn−i−1ψ dψdudv, c = σi−1σn−i−1/σn−1.
This gives
(Rαi f)(ξ) = c γn,i(α)
∫ π/2
sini−1 ψ cosα−1ψ dψ
Sn−1∩ξ⊥
Sn−1∩ξ
f(u sin ψ+vcosψ) du
ci(α)
Γ(α/2)
tα/2−1F (t) dt,
where
ci(α) =
c γn,i(α) Γ(α/2)
σi−1σn−i−1
Γ((n− α− i)/2)
2π(n−1)/2
2π(i−1)/2
as α → 0, and
F (t) = (1− t2)i/2−1
Sn−1∩ξ⊥
Sn−1∩ξ
1− t2+vt) du.
10 BORIS RUBIN
Since
Γ(α/2)
tα/2−1F (t) dt = F (0) =
Sn−1∩ξ
f(u)du = (Rif)(ξ),
we are done. �
Analytic continuation of integrals (2.9) can be realized in spherical
harmonics as Mαf=
mj,αfj,kYj,k, where
(3.4) mj,α=
(−1)j/2
Γ(j/2 + (1− α)/2)
Γ(j/2 + (n− 1 + α)/2)
if j is even,
0 if j is odd;
see [R1], [R3]. If f ∈D′(Sn−1), then Mαf is a distribution defined by
(Mαf, ω)=(f,Mαω)=
mj,α fj,k ωj,k, ω∈D(Sn−1); α 6=1, 3, 5, . . . .
Lemma 3.2. Let α, β ∈ C; α, β 6= 1, 3, 5, . . . . If α + β = 2 − n and
f ∈ De(Sn−1) (or f ∈ D′e(Sn−1)), then
(3.5) MαMβf = f.
If α, 2−n−α 6= 1, 3, 5, . . ., then Mα is an automorphism of the spaces
De(Sn−1) and D′e(Sn−1).
Proof. The equality (3.5) is equivalent to mj,αmj,β = 1, α+β = 2−n.
The latter follows from (3.4). The second statement is a consequence of
the standard theory of spherical harmonics [Ne], because the Fourier-
Laplace multiplier mj,α has a power behavior as j → ∞. �
Corollary 3.3. The Minkowski-Funk transform on the spaces De(Sn−1)
and D′e(Sn−1) can be inverted by the formula
(3.6) (M)−1 = cn−1M
2−n, cn−1 =
2π(n−2)/2
Note that there is a wide variety of diverse inversion formulas for
the Minkowski-Funk transform (see [GGG], [He], [R3] and references
therein), but all of them are, in fact, different realizations of (3.6),
depending on classes of functions.
3.2. Auxiliary statements. We establish some connections between
operator families defined above.
INTERSECTION BODIES 11
Lemma 3.4. Let α, β ∈ C; α, β 6= 1, 3, 5, . . . . If Reα > Reβ, then
Mα =MβAα,β, where Aα,β is a spherical convolution operator with the
Fourier-Laplace multiplier
(3.7) aα,β(j) =
Γ(j/2 + (1− α)/2)
Γ(j/2 + (n− 1 + α)/2)
Γ(j/2 + (n− 1 + β)/2)
Γ(j/2 + (1− β)/2)
so that aα,β(j) ∼ (j/2)β−α as j → ∞. If α and β are real numbers
satisfying α > β > 1− n, α + β < 2, then Aα,β is an integral operator
such that Aα,βf ≥ 0 for every non-negative f ∈ L1(Sn−1).
Proof. The first statement follows from (3.4). To prove the second one,
we consider integral operators
+ f)(x) =
Γ(µ/2)
(1− t2)µ/2−1(Πtf)(x) tn−νdt,(3.8)
− f)(x) =
Γ(µ/2)
(t2 − 1)µ/2−1(Π1/tf)(x) t1−νdt,(3.9)
expressed through the Poisson integral (2.1). The Fourier-Laplace mul-
tipliers of Q
+ and Q
− are
(3.10) q̂
+ (j)=
Γ((j+n−ν+1)/2)
Γ((j+n−ν+1+µ)/2)
− (j)=
Γ((j+ν−µ)/2)
Γ((j+ν)/2)
They can be easily computed by taking into account that Πt ∼ tj in
the Fourier-Laplace terms. If f ∈ L1(Sn−1) and 0 < µ < ν < n, then
integrals (3.8) and (3.9) are absolutely convergent and obey Q
± f ≥ 0
when f ≥ 0. Comparing (3.10) and (3.7), we obtain a factorization
Aα,β = Q
α−β,1−β
α−β,1−β
− (set µ = α − β, ν = 1 − β), which implies
the second statement of the lemma. �
It is convenient to introduce a special notation for the spherical
Radon transform and the generalized cosine transform with orthogonal
argument. Assuming ξ ∈ Gn,i, we denote
(3.11) (Rn−i,⊥f)(ξ) = (Rn−if)(ξ
⊥), (Rαn−i,⊥f)(ξ) = (R
n−if)(ξ
Lemma 3.5. Let f ∈ L1(Sn−1), Re α > 0; α 6= 1, 3, 5, . . . . Then
(3.12) (RiM
αf)(ξ) = c (Rα+i−1n−i,⊥ f)(ξ), ξ ∈ Gn,i, c =
2π(i−1)/2
or (replace i by n− i)
(3.13) (Rn−i,⊥M
αf)(ξ) =
2π(n−i−1)/2
σn−i−1
(Rα+n−i−1i f)(ξ).
If f ∈ De(Sn−1), then (3.12) and (3.13) extend to Reα ≤ 0 by analytic
continuation.
12 BORIS RUBIN
Proof. For Reα > 0,
αf)(ξ) = γn(α)
Sn−1∩ξ
f(θ)|θ · u|α−1 dθ.
Since |θ · u| = |Prξθ||vθ · u| for some vθ ∈ Sn−1 ∩ ξ, by changing the
order of integration, we obtain
αf)(ξ) = γn(α)
f(θ)|Prξθ|α−1 dθ
Sn−1∩ξ
|vθ · u|α−1dξu.
The inner integral is independent on vθ and can be easily evaluated:
Sn−1∩ξ
|vθ · u|α−1dξu =
|t|α−1(1− t2)(i−3)/2 dt
2π(i−1)/2 Γ(α/2)
σi−1 Γ((i+ α− 1)/2)
This implies (3.12). �
The following statement is dual to Lemma 3.5.
Lemma 3.6. Let µ ∈ M(Gn,i), α 6= 1, 3, 5, . . . . Then
(3.14) MαR∗iµ = c
Rα+i−1n−i µ
⊥, c = 2π(i−1)/2/σi−1,
in the D′(Sn−1)-sense. If Reα > 0 and µ is absolutely continuous with
density ϕ ∈ L1(Gn,i), then
(3.15) MαR∗iϕ = c
Rα+i−1n−i ϕ
almost everywhere on Sn−1. If ϕ ∈ D(Gn,i), then (3.15) extends to all
complex α 6= 1, 3, 5, . . . by analytic continuation.
Proof. Let ω ∈ De(Sn−1) (it suffices to consider only even test func-
tions). By (2.4) and (3.12),
(MαR∗iµ, ω) = (µ,RiM
αω) = c (µ,Rα+i−1n−i,⊥ ω) = c (µ
⊥, Rα+i−1n−i ω).
This gives the result. �
The next statement contains explicit representations of the right in-
verse of the dual Radon transform R∗i (note that R
i is non-injective on
D(Gn,i) when 1 < i < n− 1).
Lemma 3.7. Every function f ∈De(Sn−1) is represented as f=R∗iAf ,
where A : De(Sn−1) → D(Gn,i),
(3.16) Af = c1R
i f = c2Rn−i,⊥M
2−nf,
π(1−i)/2σn−2
σn−i−1
Γ((n− i)/2)
Γ((n− 1)/2)
, c2 =
2πn/2−1
INTERSECTION BODIES 13
Proof. The coincidence of expressions in (3.16) follows from (3.13). To
prove the first equality, we invoke spherical convolutions defined by
analytic continuation of the integral
(3.17) (Qαf)(θ)=
σn−1Γ((n−1−α)/2)
2π(n−1)/2Γ(α/2)
(1−|u·θ|2)(α−n+1)/2f(u)du,
Reα > 0, α − n 6= 0, 2, 4, . . . , so that Q0f = f [R2]. By Theorem
1.1 from [R2], R∗iR
i f = c
α+i−1f , and therefore (set α = 1 − i),
i f = c
1 f , as desired. �
The next statement provides an intriguing factorization of the Minkowski-
Funk transform in terms of Radon transforms associated to mutually
orthogonal subspaces. This factorization can be useful in different oc-
currences.
Theorem 3.8. For f ∈ L1(Sn−1) and 0 < i < n,
(3.18) Mf = R∗iRn−i,⊥f.
Proof. By (2.3),
(R∗iRn−i,⊥f)(θ) =
SO(n−1)
(Rn−i,⊥f)(rθγR
i) dγ
SO(n−1)
(Rn−if)(rθγR
n−i) dγ
SO(n−1)
Sn−1∩rθγR
f(v) dv
Sn−1∩Rn−i
SO(n−1)
f(rθγw) dγ.
The inner integral is independent on w ∈ Sn−1 ∩ Rn−i and equals
(Mf)(θ). This gives (3.18). �
3.3. Restriction theorems. Theorems of such type deal with traces
of functions on lower dimensional subspaces and are well known, for in-
stance, in the theory of function spaces. To the best of our knowledge,
traces of functions represented by Radon transforms or, more generally,
by the generalized cosine transforms , were not studied systematically
and deserve particular attention, because they provide analytic back-
ground to a series of results related to sections of star bodies; cf. [R3,
Sec. 3.5], [FGW]. Given a subspace η ∈ Gn,m and k < m, we denote
by Gk(η) the manifold of all k-dimensional subspaces of η.
14 BORIS RUBIN
Theorem 3.9. Let f ∈ Ce(Sn−1), 1 ≤ k < m < n, λ 6= 0,−2,−4, . . . .
If Reλ < k, then for every η ∈ Gn,m and every ξ ∈ Gk(η),
(3.19) (Rk−λn−kf)(ξ
⊥) = (Rk−λm−kT
η f)(ξ
⊥ ∩ η),
where
(3.20) (T λη f)(u) = c̃
Sn−1∩(η⊥⊕Ru)
f(w)|u · w|m−λ−1 dw,
u ∈ Sn−1 ∩ η, c̃ = π(m−n)/2 σn−m/2.
In particular (let λ→ k),
(3.21) (Rn−kf)(ξ
⊥)=c (Rm−kT
η f)(ξ
⊥ ∩ η), c= π
(n−m)/2 σm−k−1
σn−k−1
Proof. By (2.6),
(Rk−λn−kf)(ξ
⊥)=γn,n−k(k − λ)
|Prξθ|−λ f(θ) dθ.
We represent θ in bi-spherical coordinates as
(3.22) θ = ucosψ + v sinψ,
where
u ∈ Sn−1 ∩ η ∼ Sm−1, v ∈ Sn−1 ∩ η⊥ ∼ Sn−m−1, 0 ≤ ψ ≤ π/2,
dθ = c′′ sinn−m−1 ψ cosm−1ψ dψdudv, c′′ = σm−1σn−m−1/σn−1.
If ξ ⊂ η, then |Prξθ| = |Prξ[Prηθ]| = |Prξu| cosψ, and therefore,
(Rk−λn−kf)(ξ
⊥) = γm,m−k(k − λ)
Sn−1∩η
|Prξu|−λ(T λη f)(u) du,
where
(T λη f)(u) =
c′′ γn,n−k(k − λ)
γm,m−k(k − λ)
∫ π/2
sinn−m−1 ψ cosm−λ−1ψ dψ
Sn−1∩η⊥
f(ucosψ+v sin ψ) dv
π(m−n)/2 σn−m
Sn−1∩(η⊥⊕Ru)
f(w)|u · w|m−λ−1 dw.
Formula (3.21) follows from (3.19) by (3.1). �
INTERSECTION BODIES 15
Theorem 3.10. Let f ∈De(Sn−1), η∈Gn,m, 1<m<n. Suppose that
f =M1−λg, where Reλ < m, λ 6= 0,−2,−4, . . . . Then the restriction
of f onto η is represented as f =M1−λ
Sn−1∩η
T λη g, where T
η has the form
(3.20) andM1−λ
Sn−1∩η
denotes the same operator M1−λ, but on the sphere
Sn−1 ∩ η.
Proof. For Reλ < 1, the statement is a particular case of Theorem
3.9 (set k = 1). For other values of λ, the result follows by analytic
continuation. �
Remark 3.11. The restriction λ 6= 0,−2,−4, . . . in Theorems 3.9 and
3.10 is caused by the Gamma function Γ(λ/2) in the numerator of the
corresponding normalizing factor. It is evident from the proof, that
both theorems remain true also for λ = −2ℓ, ℓ ∈ N, if we remove the
normalizing factor. Then M1−λ in Theorem 3.10 will be substituted
for M̃1+2ℓ; see (2.11).
We will need the following generalization of Theorem 3.10.
Theorem 3.12. Let f ∈Ce(Sn−1), µ ∈ Me+(Sn−1), and let η∈Gn,m,
1<m<n. Suppose that f =M1−λµ, if λ < m, λ 6= −2ℓ, ℓ ∈ N, and
f=M̃1+2ℓµ, if λ = −2ℓ.
(i) There is a measure ν ∈ Me+(Sn−1 ∩ η) such that the restriction of
f onto Sn−1 ∩ η is represented as f =M1−λ
Sn−1∩η
(ii) If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then (i) holds with dν(θ) =
(T λη g)(θ)dθ, where T
η g has the form (3.20).
(iii) If λ = −2ℓ, ℓ ∈ N, then (i) and (ii) hold with M1−λ
Sn−1∩η
substituted
for M̃1+2ℓ
Sn−1∩η
Proof. STEP 1. Let first λ < m, λ 6= 0,−2,−4, . . . . We invoke the
Poisson integral (2.1) so that
Πtf = ΠtM
1−λµ =M1−λgt, gt = Πtµ ∈ De(Sn−1), t ∈ (0, 1).
Since f is continuous, then Πtf converges to f as t → 0 uniformly
on Sn−1, and therefore, uniformly on Sn−1 ∩ η. Hence, for any test
function ω ∈ D(Sn−1 ∩ η), owing to Theorem 3.10, we have
(f, ω) = lim
(Πtf, ω) = lim
(M1−λgt, ω)
= lim
(M1−λ
Sn−1∩η
T λη gt, ω) = lim
(T λη gt,M
Sn−1∩η
= lim
(νt,M
Sn−1∩η
ω), νt = T
η gt.(3.23)
Thus, lim
(νt,M
Sn−1∩η
ω) exists for every ω∈D(Sn−1∩η). If ω is even,
i.e., ω ∈ De(Sn−1 ∩ η), then, by Lemma 3.2, we can replace ω by
16 BORIS RUBIN
M1−m+λ
Sn−1∩η
ω and conclude that the limit lim
(νt, ω) is well-defined for every
ω ∈ De(Sn−1∩η). Since νt = T λη Πtµ is an even function and the generic
test function ω ∈ D(Sn−1 ∩ η) can be represented as ω+ + ω−, where
ω± are even and odd, respectively, it follows that the limit lim
(νt, ω) =
(νt, ω+) is well-defined for every ω ∈ D(Sn−1∩ η) (not only for even
ω, as stated above). Since D′(Sn−1∩ η) is weakly complete, there is an
even distribution ν in D′(Sn−1 ∩ η) so that
(ν, ω) = lim
(νt, ω), ω ∈ D(Sn−1 ∩ η).
Furthermore, since (νt, ω) = (T
η Πtµ, ω) is non-negative for every non-
negative ω ∈ D(Sn−1 ∩ η) and every t ∈ (0, 1), then ν is a positive
distribution and, by Theorem 9.1, ν is a measure in Me+(Sn−1 ∩ η).
Thus, by (3.23), (f, ω) = lim
(νt,M
Sn−1∩η
ω) = (ν,M1−λ
Sn−1∩η
ω), which
means that f =M1−λ
Sn−1∩η
ν, as desired.
If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then νt = T λη Πtg tends to T λη g uni-
formly on Sn−1∩η as t→ 0. Hence, by (3.23), (f, ω) = (T λη g,M1−λSn−1∩ηω),
which means f =M1−λ
Sn−1∩η
T λη g.
STEP 2. Consider the case λ = −2ℓ, ℓ ∈ N, when f = M̃1+2ℓµ,
µ ∈ Me+(Sn−1), and the operator T λη = T−2ℓη has the form
(T−2ℓη h)(u) = c̃
Sn−1∩(η⊥⊕Ru)
|u · w|m+2ℓ−1 h(w) dw,
cf. (3.20). For any functions h ∈ C(Sn−1) and ω ∈ C(Sn−1 ∩ η),
(3.24) (T−2ℓη h, ω) = (h,
T−2ℓη ω),
where
T−2ℓη ω)(θ)=
Γ(m/2)
2Γ(n/2)
( Prηθ
|Prηθ|
|Prηθ|2ℓ ∈ C(Sn−1).
INTERSECTION BODIES 17
Indeed, using bi-spherical coordinates (see (3.22)), we have
(T−2ℓη h, ω) = c̃
Sn−1∩η
ω(u)du
Sn−1∩(η⊥⊕Ru)
h(w)|u · w|m+2ℓ−1 dw
c̃ σn−m−1
Sn−1∩η
ω(u)du
sinn−m−1 ψ cosm+2ℓ−1ψ dψ
Sn−1∩η⊥
h(ucosψ+v sin ψ) dv
c̃ σn−m−1
c′′ σn−m
h(θ)ω
( Prηθ
|Prηθ|
|Prηθ|2ℓ dθ = (h,
T−2ℓη ω).
Let h = Πtµ and observe that the limit lim
(T−2ℓη Πtµ, ω) exists, be-
cause, by (3.24), (T−2ℓη Πtµ, ω) = (Πtµ,
T−2ℓη ω) → (µ,
T−2ℓη ω). Note
that (T−2ℓη Πtµ, ω) ≥ 0 for any non-negative ω ∈ C(Sn−1∩η). Applying
the standard completeness argument (as in Step 1), we conclude, that
there is a measure ν ∈ M+(Sn−1 ∩ η) such that
(T−2ℓη Πtµ, ω) = (ν, ω) ∀ω ∈ C(Sn−1 ∩ η).
Using this equality, for f = M̃1+2ℓµ we obtain
(f, ω) = lim
(Πtf, ω) = lim
(ΠtM̃
1+2ℓµ, ω) = lim
(M̃1+2ℓΠtµ, ω)
(use Theorem 3.10 and Remark 3.11)
= lim
(M̃1+2ℓ
Sn−1∩η
T−2ℓη Πtµ, ω) = lim
(T−2ℓη Πtµ, M̃
Sn−1∩η
= (ν, M̃1+2ℓ
Sn−1∩η
This gives the result.
If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then, by Theorem 3.10 and Remark
3.11, for θ ∈ Sn−1 ∩ η we have
(Πtf)(θ) = (ΠtM̃
1+2ℓg)(θ) = (M̃1+2ℓΠtg)(θ) = (M̃
Sn−1∩η
T−2ℓη Πtg)(θ).
Owing to continuity of the operators M̃1+2ℓ
Sn−1∩η
, T−2ℓη , and Πt in the
relevant spaces of continuous functions, by passing to the limit as t→ 0,
we obtain f(θ) = (M̃1+2ℓ
Sn−1∩η
T−2ℓη g)(θ), θ ∈ Sn−1 ∩ η, as desired. �
18 BORIS RUBIN
4. Positive Definite Homogeneous Distributions
We remind some known facts; see, e.g., [GS], [Le]. Let S(Rn) be the
Schwartz space of rapidly decreasing C∞-functions on Rn and S ′(Rn)
its dual. The Fourier transform of F ∈ S ′(Rn) is defined by
〈F̂ , φ̂〉 = (2π)n〈F, φ〉, φ̂(y) =
φ(x) eix·y dx, φ ∈ S(Rn).
A distribution F ∈ S ′(Rn) is homogeneous of degree λ ∈ C if for any
φ ∈ S(Rn) and any a > 0, 〈F, φ(x/a)〉 = aλ+n 〈F, φ〉. Homogeneous dis-
tributions on Rn are intimately connected with distributions on Sn−1.
Let first f ∈ L1(Sn−1), (Eλf)(x) = |x|λf(x/|x|), x ∈ Rn \ {0}. The
operator Eλ generates a meromorphic S ′-distribution
〈Eλf, φ〉= a.c.
rλ+n−1u(r)dr, u(r) =
f(θ)φ(rθ)dθ,
where “a.c.” denotes analytic continuation in the λ-variable. The dis-
tribution Eλf is regular if Reλ > −n and admits simple poles at
λ = −n,−n − 1, . . .. The above definition extends to all distributions
f ∈ D′(Sn−1) by the formula
〈Eλf, φ〉 = a.c.
rλ+n−1u(r)dr, u(r) = (f, φ(rθ)), 1
and the map Eλ : D′(Sn−1) → S ′(Rn) is weakly continuous. If f
is orthogonal to all spherical harmonics of degree j, then the deriv-
ative u(j)(r) equals zero at r = 0 and the pole at λ = −n − j is
removable. In particular, if f is an even distribution, i.e., (f, ϕ) =
(f, ϕ−), ϕ−(θ) = ϕ(−θ) ∀ϕ ∈ D(Sn−1), then the only possible poles
of Eλf are −n,−n− 2,−n− 4, . . . .
The Fourier transform of homogeneous distributions was extensively
studied by many authors; see [Sa3] and references therein. We restrict
our consideration to even distributions, when the operator family {Mα}
defined by (2.9) naturally arises thanks to the formula
(4.1) [E1−n−αf ]
∧ = 21−απn/2Eα−1M
This formula amounts to Semyanistyi [Se]. If f ∈ De(Sn−1), then (4.1)
holds pointwise for 0 < Reα < 1 (see, e.g., Lemma 3.3 in [R1] ) and
extends in the S ′-sense to all α ∈ C satisfying
(4.2) α /∈ {1, 3, 5, . . .} ∪ {1− n,−n− 1,−n− 3, . . .}.
1Here and on, different notations 〈·, ·〉 and (·, ·) are used for distributions on Rn
and Sn−1, respectively.
INTERSECTION BODIES 19
Since De(Sn−1) is dense in D′e(Sn−1) and the maps E1−n−α and Eα−1
are weakly continuous from D′e(Sn−1) to S ′(Rn), then (4.1) extends to
all f ∈ D′e(Sn−1).
Regarding the cases excluded in (4.2), we note that if α = 1+ 2ℓ for
some ℓ = 0, 1, . . ., then (4.1) is meaningful if and only if f is orthogonal
to all spherical harmonics of degree 2ℓ. If α = 1 − n − 2ℓ for some
ℓ = 0, 1, . . ., then, according to the spherical harmonic decomposition
j,k fj,kYj,k, j even, formula (4.1) is substituted for the following:
[E2ℓf ]
∧(ξ) = (2π)n
fj,k(−∆)ℓ−j/2Yj,k(i∂) δ(ξ)(4.3)
+2n+2ℓπn/2E−n−2ℓM
1−n−2ℓ
fj,kYj,k
where −∆ is the Laplace operator, ∂ = (∂/∂ξ1, . . . , ∂/∂ξn), and δ(ξ)
is the delta function. It is worth noting that for α = 1, 3, 5, . . ., the
distribution [E1−n−αf ]
∧ can also be understood in the regularized sense
without any orthogonality assumptions. However, such regularization
does not preserve homogeneity; see [Sa1], [Sa3].
Our main concern is positivity and positive definiteness of even ho-
mogeneous distributions. The reader is referred to [GV] for the general
theory. A distribution F ∈ S ′(Rn) is positive if 〈F, φ〉 ≥ 0 for all non-
negative φ ∈ S(Rn). A similar definition holds for distributions on the
sphere and on Rn \ {0}. A distribution F ∈ S ′(Rn) is positive definite
if F̂ is positive. For our purposes, it is important to know, which even
homogeneous distributions are positive definite. Let us rewrite (4.1)
and (4.2) with 1− n− α replaced by −λ. We have
(4.4) [E−λf ]
∧ = 2n−λπn/2Eλ−nM
1+λ−nf,
(4.5) λ /∈ Λ0, Λ0 = {n, n + 2, n+ 4 . . .} ∪ {0,−2,−4, . . .}.
Theorem 4.1. Let λ ∈ R \ Λ0, f ∈ D′e(Sn−1).
(i) If λ < 0 and E−λf is a positive definite distribution, then f = 0.
(ii) For all λ ∈ R \ Λ0, the following statements are equivalent:
(a) [E−λf ]
∧ is a positive distribution on Rn \{0} (for λ > 0, this can
be replaced by “E−λf is a positive definite distribution on R
(b) M1+λ−nf ∈ Me+(Sn−1);
(c) f =M1−λµ for some measure µ ∈ Me+(Sn−1).
Furthermore, for any real λ 6= 0,−2,−4, . . ., and any i = 1, 2, . . . , n−1,
(c) is equivalent to
(d) Rif = R
n−i,⊥µ for some measure µ ∈ Me+(Sn−1).
20 BORIS RUBIN
Proof. (i) Choose φ(x) = exp(−|x|m) pt,θ(x/|x|), where m ∈ 2N and
pt,θ(·) is the Poisson kernel
(4.6) pt,θ(u) =
1− t2
(1− 2tu · θ + t2)n/2
, 0 < t < 1; u, θ ∈ Sn−1.
Then 〈Eλ−nM1+λ−nf, φ〉 = cλ(ΠtM1+λ−nf)(θ), where
cλ = a.c.
rλ−1 exp(−rm) dr = m−1Γ(λ/m)
and (ΠtM
1+λ−nf)(θ) is the Poisson integral of M1+λ−nf . If E−λf is a
positive definite distribution, then, by (4.4), Eλ−nM
1+λ−nf is a positive
distribution. On the other hand, if λ < 0 and m > −λ, then cλ < 0.
Hence 〈Eλ−nM1+λ−nf, φ〉 can be non-negative for every non-negative
φ ∈ S(Rn) only if (ΠtM1+λ−nf)(θ) = 0 for every 0 < t < 1 and
θ ∈ Sn−1. The latter implies M1+λ−nf = 0, which is equivalent to
f = 0 because M1+λ−n is injective; see Lemma 3.2.
(ii) Let [E−λf ]
∧ be a positive distribution on Rn \{0}. It means that
for every φ ∈ S(Rn) such that φ ≥ 0 and 0 /∈ suppφ, 〈[E−λf ]∧, φ〉 ≥ 0
or, by (4.4), 〈Eλ−nM1+λ−nf, φ〉 ≥ 0. Choose φ(x) = ψ(|x|)ω(x/|x|),
where ω ∈ D(Sn−1), ω ≥ 0, and ψ is a smooth non-negative function
such that
rα+n−2ψ(r)dr = 1 and 0 /∈ suppψ. Then
〈Eλ−nM1+λ−nf, φ〉 = (M1+λ−nf, ω) ≥ 0,
and therefore, M1+λ−nf ∈ Me+(Sn−1); see Theorem 9.1.
Conversely, let µ = M1+λ−nf ∈ Me+(Sn−1) and let φ ∈ S(Rn);
φ ≥ 0. In the case λ < 0 we additionally assume 0 /∈ suppφ. By (4.4),
〈[E−λf ]∧, φ〉 = 2n−λπn/2 〈Eλ−nµ, φ〉
= 2n−λπn/2
rλ−1dr
φ(rθ)dµ(θ) ≥ 0.
This proves equivalence of (a) and (b). Equivalence of (b) and (c)
follows from Lemma 3.2.
Let us prove the equivalence of (c) and (d). If Rif = R
n−i,⊥µ,
µ ∈ Me+(Sn−1), then, by (3.15),
(f, R∗iϕ) = (Rif, ϕ) = (R
n−i,⊥µ, ϕ) = (µ,
Ri−λn−iϕ
= c−1(µ,M1−λR∗iϕ), ϕ ∈ D(Gn,i).
Since any function ω ∈ De(Sn−1) can be expressed as ω = R∗iϕ for
some ϕ ∈ D(Gn,i) (see Lemma 3.7), this gives (f, ω) = c−1(µ,M1−λ, ω)
which is (c). Conversely, let f = M1−λµ, µ ∈ Me+(Sn−1), that is,
(f, ω) = (µ,M1−λ, ω) for every ω ∈ De(Sn−1). Choose ω = R∗iϕ, ϕ ∈
INTERSECTION BODIES 21
D(Gn,i). Then, as above, (f, R∗iϕ) = (µ,M1−λR∗iϕ) = c (µ,
Ri−λn−iϕ
which gives (d). �
5. λ-intersection bodies
5.1. Definitions and comments. We remind that Kn is the set of
all origin-symmetric star bodies K in Rn, n ≥ 2; ρK and || · ||K are the
radial function and the Minkowski functional of K. The following defi-
nitions and statements are motivated by Theorem 4.1 and the previous
consideration. Let λ be a real number,
(5.1) sλ =
1 if λ > 0, λ 6= n, n+ 2, n+ 4, . . . ,
Γ(λ/2) if λ < 0, λ 6= −2,−4, . . . .
The values λ = 0, n, n + 2, n + 4, . . . will not be considered in the
following, but values λ = −2,−4, . . . will be included. They become
meaningful if we change normalization. For λ 6= 0, n, n + 2, n + 4 . . . ,
let Inλ be the set of bodies K ∈ Kn, for which there is a measure
µ ∈ Me+(Sn−1) such that sλρK = M1−λµ if λ 6= −2ℓ, ℓ ∈ N, and
ρK = M̃
1−λµ ≡ M̃1+2ℓµ, otherwise. The equality sλρK = M1−λµ
means that for any ϕ ∈ D(Sn−1),
ρkK(θ)ϕ(θ) dθ =
(M1−λϕ)(θ) dµ(θ),
where for λ ≥ 1, (M1−λϕ)(θ) is understood in the sense of analytic
continuation. We remind the notation
Λ0 = {n, n+ 2, n+ 4 . . .} ∪ {0,−2,−4, . . .}.
Theorem 5.1. For λ ∈ R\Λ0, the following statements are equivalent:
(a) K ∈ Inλ ;
(b) The Fourier transform [sλ || · ||−λK ]∧ is a positive distribution on
n\{0} (for λ > 0, this can be replaced by “|| · ||−λK is a positive definite
distribution on Rn”);
(c) sλM
1+λ−nρλK ∈ Me+(Sn−1);
The theorem is an immediate consequence of Theorem 4.1 if the lat-
ter is applied to f = sλρ
K . Another useful characterization is provided
by Theorem 4.1 (d).
Theorem 5.2. Let λ ∈ R \ Λ0. If K ∈ Inλ , then for every i ∈
{1, 2, . . . , n−1} there is a measure µ ∈ Me+(Sn−1) such that sλRiρλK =
Ri−λn−i,⊥µ. Conversely, if
sλRiρ
K = R
n−i,⊥µ
for some i ∈ {1, 2, . . . , n−1} and some µ ∈ Me+(Sn−1), then K ∈ Inλ .
22 BORIS RUBIN
Although Inλ was called “the set of bodies”, the definition of this set
is purely analytic and extra work is needed to understand what bodies
(if any) actually constitute the class Inλ .
The following comments will be helpful.
1. The case λ > n is not so interesting, because by Theorem 5.1(c),
Inλ is either empty (if Γ((n − λ)/2) < 0) or coincides with the whole
class Kn (if Γ((n− λ)/2) > 0).
2. The case λ ∈ (0, n) agrees with the concept of isometric embed-
ding of the space (Rn, || · ||K) into L−p, p = λ; see Introduction. In the
framework of this concept, all bodies K ∈ Inλ can be regarded as “unit
balls of n-dimensional subspaces of L−λ”.
3. If K ∈ Inλ , where λ < 0 (one can replace λ by = −p, p > 0), then
||u||pK =
|θ · u|p dµ(θ)
for some µ ∈ Me+(Sn−1). This is the well known Lévy representation,
characterizing isometric embedding of the space (Rn, || · ||K) into Lp;
see Lemma 6.4 in [K4]. Statement (b) in Theorem 5.1 agrees with
Theorem 1.9. Keeping this terminology, we can state the following
Proposition 5.3. Let p > −n, p 6= 0. Then (Rn, || · ||K) embeds
isometrically in Lp if and only if K ∈ In−p.
4. If λ = k ∈ {1, 2, . . . , n−1}, then Inλ = Ink coincides with the class
of k-intersection bodies; see Definition 1.7 and Theorem 1.8. Theorems
5.1 and 5.2 provide new characterizations of this class.
These comments inspire the following
Definition 5.4. Let λ < n, λ 6= 0. A body K ∈ Kn is said to be a
λ-intersection body if K ∈ Inλ , or, in other words, if there is a measure
µ ∈ Me+(Sn−1) such that sλρλK = M1−λµ if λ 6= −2ℓ, ℓ ∈ N, and
ρ−2ℓK = M̃
1+2ℓµ, otherwise.
The result of Theorem 5.2 for λ = i = k can serve as an alternative
definition of k-intersection bodies in terms of Radon transforms. This
definition agrees with Definition 1.6 and mimics Definition 1.2.
Definition 5.5. Let k ∈ {1, 2, . . . , n − 1}. A body K ∈ Kn is a k-
intersection body if there is a non-negative measure µ on Sn−1 such
(5.2) (Rkρ
K)(ξ) = (Rn−kµ)(ξ
⊥), ξ ∈ Gn,k.
INTERSECTION BODIES 23
Equality (5.2) is understood in the weak sense according (2.5). Namely,
for ϕ ∈ C(Gn,k) and ϕ⊥(η) = ϕ(η⊥), η ∈ Gn,n−k, (5.2) means
(5.3)
K)(ξ)ϕ(ξ) dξ =
(R∗n−kϕ
⊥)(θ) dµ(θ).
5.2. λ-intersection bodies of star bodies and closure in the ra-
dial metric. As we mentioned in Introduction, the class of intersection
bodies, which coincides with Inλ when λ = 1, is the closure in the ra-
dial metric of the class of intersection bodies of star bodies. Below
we extend this result to all λ < n, λ 6= 0, in the framework of the
unique approach. We remind (see Definition 1.6) that K ∈ Kn is a
k-intersection body of a body L ∈ Kn and write K = IBk(L) if
(5.4) volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) ∀ξ ∈ Gn,k.
Let IBk,n be the set of all bodies K ∈ Kn satisfying (5.4) for some
L ∈ Kn.
How can we extend the purely geometric property (5.4) to non-
integer values of k? To this end, we first express (5.4) in terms of
the generalized cosine transforms (2.9).
Lemma 5.6. If K = IBk(L) is infinitely smooth, then
(5.5) ρn−kL =cM
1−n+kρkK , ρ
−1M1−kρn−kL ,
c = πk−n/2(n− k)/k.
Proof. We make use of (3.13), where we set i = k, α = 1 − n + k and
f = ρkK . By (3.1), this gives
(5.6) Rkρ
K = c̃Rn−k,⊥M
1−n+kρkK , c̃ =
πk−n/2 σn−k−1
On the other hand, if K = IBk(L) is infinitely smooth, then, according
to (5.4) and the equality
(5.7) volk(K ∩ ξ) =
K)(ξ),
we have
(5.8) Rkρ
k σn−k−1
(n− k) σk−1
Rn−k,⊥ρ
Comparing (5.6) and (5.8), owing to injectivity of the Radon transform,
we obtain the first equality in (5.5). The second equality follows from
the first one by (3.5). �
24 BORIS RUBIN
Equalities (5.5) are extendable to non-integer values of k. We denote
cλ,n = π
λ−n/2(n−λ)/λ,
and let sλ be defined by (5.1).
Definition 5.7. Let λ < n, λ 6= 0; K,L ∈ Kn. We say that K is a
λ-intersection body of L and write K = IBλ(L) if sλρλK=c−1λ,nM1−λρ
in the case λ 6= −2ℓ, ℓ ∈ N, and ρ−2ℓK = M̃1+2ℓρ
L , otherwise. The
set of all λ-intersection bodies of star bodies will be denoted by IBλ,n.
We also denote
(5.9) IB∞λ,n={K ∈ IBλ,n : ρK ∈ De(Sn−1)}.
By (3.5), equality sλρ
K = c
1−λρn−λL is equivalent to ρ
sλ cλ,nM
1−n+λρλK . Both equalities are generally understood in the sense
of distributions, for instance,
K , ϕ) = c
λ,n(ρ
1−λϕ), ϕ ∈ D(Sn−1).
If K (or L) is smooth, then sλρ
K(θ)=c
λ,n(M
1−λρn−λL )(θ) pointwise for
every θ∈Sn−1.
Theorem 5.8. Let λ < n, λ 6= 0. If λ 6= −2ℓ, ℓ ∈ N, then the class
Inλ of λ-intersection bodies is the closure of the classes IBλ,n and IB∞λ,n
of λ-intersection bodies of star bodies in the radial metric:
(5.10) Inλ = cl IBλ,n = cl IB∞λ,n.
If λ = −2ℓ, ℓ ∈ N, then Inλ ⊂ cl IBλ,n = cl IB∞λ,n.
Proof. STEP 1. We first prove that Inλ ⊂ cl IB∞λ,n. Let K ∈ Inλ , i.e.,
(a) sλρ
1−λµ, µ ∈ Me+(Sn−1), if λ 6= −2ℓ, ℓ ∈ N, and
(b) ρ−2ℓK = M̃
1+2ℓµ, otherwise.
Our aim is to define a sequence Kj ∈ IB∞λ,n such that ρKj → ρK in the
C-norm. Consider the Poisson integral Πtρ
K (see (2.1)), that converges
to ρλK in the C-norm when t→ 1. In the case (a), for any test function
ω ∈ D(Sn−1) we have
K , ω) = (ρ
K ,Πtω) = s
λ (µ,M
1−λΠtω) = s
1−λΠtµ, ω).
Similarly, in the case (b), we have a pointwise equality (Πtρ
K )(θ) =
(M̃1+2ℓΠtµ)(θ), θ ∈ Sn−1. Choose Kj so that ρλKj = Πtjρ
K , where tj is
a sequence in (0, 1) approaching 1. Clearly, Kj converges to K in the
radial metric. Moreover, Kj ∈ IB∞λ,n, because ρλKj = s
1−λρn−λLj
and ρ−2ℓKj = M̃
1+2ℓρn+2ℓLj , where the bodies Lj are defined by ρ
cλ,nΠtjµ in the case (a), and ρ
= Πtjµ in the case (b), respectively.
INTERSECTION BODIES 25
Conversely, let K ∈ cl IB∞λ,n, λ 6= −2,−4, . . . . It means that there
is a sequence of Kj ∈ IB∞λ,n such that lim
||ρK − ρKj ||C(Sn−1) = 0 and
= c−1λ,nM
1−λρn−λLj , ρLj ∈ De+(S
n−1). If j → ∞, then for every
ω ∈ D(Sn−1),
(5.11) sλ(ρ
,M1−n+λω) → sλ(ρλK ,M1−n+λω)=sλ(M1−n+λρλK , ω).
The right-hand side of (5.11) is non-negative, because by (3.5), for
every j and every ω ∈ De+(Sn−1),
,M1−n+λω) = c−1λ,n(M
1−λρn−λLj ,M
1−n+λω) = c−1λ,n(ρ
, ω) ≥ 0.
By Theorem 9.1, it follows that sλM
1−n+λρλK is a non-negative mea-
sure. We denote it by µ. By (3.5), for any ω ∈ D(Sn−1),
K , ω) = sλ(M
1−n+λρλK ,M
1−λω) = (µ,M1−λω) = (M1−λµ, ω),
i.e., K∈Inλ . This gives IB∞λ,n⊂Inλ and, by above, Inλ =cl IB∞λ,n.
STEP 2. It remains to prove that cl IB∞λ,n = cl IBλ,n. Since IB∞λ,n ⊂
IBλ,n, then cl IB∞λ,n ⊂ cl IBλ,n. To prove the opposite inclusion, let
K ∈ cl IBλ,n and consider the case λ 6= −2,−4, . . . . We have to show
that there is a sequence of smooth bodies Kj, which converges to K in
the radial metric and such that sλρ
= c−1λ,nM
1−λρn−λLj for some bodies
Lj ∈ Kn. Since K ∈ cl IBλ,n, there is a sequence K̃j ∈ Kn such that
||ρK̃j −ρK ||C(Sn−1) = 0 and sλρ
= c−1λ,nM
1−λρn−λ
for some bodies
L̃j ∈ Kn. We define smooth bodies Kj and Lj by
ρλKj = Π1−1/jρ
, ρn−λLj = Π1−1/jρ
where Π1−1/j stands for the Poisson integral with parameter 1 − 1/j.
Since operators Π1−1/j andM
1−λ commute, then sλρ
=c−1λ,nM
1−λρn−λLj ,
and therefore, Kj ∈ IB∞λ,n. On the other hand, by the properties of the
Poisson integral [SW],
|ρλKj − ρ
K | ≤ |Π1−1/jρλK̃j − Π1−1/jρ
K |+ |Π1−1/jρλK − ρλK | → 0
as j → ∞. It means, that K ∈ cl IB∞λ,n or cl IBλ,n ⊂ cl IB∞λ,n. Hence,
by above, cl IBλ,n = cl IB∞λ,n. For λ = −2,−4, . . . , the argument is
similar. �
Remark 5.9. If λ = −2,−4, . . . , we cannot prove the coincidence of
Inλ and cl IB∞λ,n, because the proof of the embedding cl IB∞λ,n ⊂ Inλ
relies heavily on the fact that M1−λ is an isomorphism of De(Sn−1). If
λ = −2,−4, . . . , this is not so, and the operator M̃1−λ has a nontrivial
kernel, which consists of spherical harmonics of degree > 2ℓ; see [R1]
for details.
26 BORIS RUBIN
It is interesting to translate Theorem 5.8 for λ = −p, p > 0, into
the language of isometric embeddings. Ignoring a non-important pos-
itive constant factor and using polar coordinates, one can replace the
equalities sλρ
K = c
1−λρn−λL and ρ
K = M̃
1+2ℓρn+2ℓL in Definition
5.7 by
(5.12) ||u||pK =
|x · u|p dx, u ∈ Sn−1.
Corollary 5.10.
(i) A unit ball of every n-dimensional subspace of Lp, can be approxi-
mated in the radial metric by bodies K, defined by (5.12), where L ∈ Kn
has a C∞ boundary.
(ii) If, moreover, p 6= 2, 4, . . . , then the set of unit balls of all n-
dimensional subspaces of Lp, can be identified with the closure in the
radial metric of the set of bodies K satisfying (5.12) for some smooth
body L ∈ Kn (one can also consider arbitrary bodies L ∈ Kn).
5.3. Central sections of λ-intersection bodies. It is known, that
a cross-section K ∩ η of a body K ∈ Ink by any m-dimensional central
plane η is a k-intersection body in η provided 1 ≤ k < m < n. This
fact was established in [Mi1, Proposition 3.17] by using Theorem 1.8
and a certain approximation procedure. Below we present more general
results, including sections of k-intersection bodies of star bodies and
the case of non-integer k = λ. These results are consequences of the
restriction theorems from Section 3.3.
Theorem 5.11. Let 1 ≤ k < m < n, η ∈ Gn,m. If K = IBk(L) in Rn,
then K ∩ η = IBk(L̃) in η, where the body L̃ is defined by
(5.13) ρm−k
(u) = ck,m,n
Sn−1∩(η⊥⊕Ru)
ρn−kL (w)|u · w|
m−k−1 dw,
u ∈ Sn−1 ∩ η, ck,m,n =
(m− k) σn−m
2(n− k)
Proof. By (5.7) and (3.21) (with f = ρn−kL ),
volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) =
σn−k−1
(Rn−kρ
L )(ξ
c σn−k−1
(Rm−kT
L )(ξ
⊥ ∩ η)(5.14)
σm−k−1
(Rm−kρ
)(ξ⊥ ∩ η) = volm−k(L̃ ∩ ξ⊥),
as desired. �
INTERSECTION BODIES 27
Theorem 5.11 has the following generalization.
Theorem 5.12. Let 1 < m < n, η ∈ Gn,m and suppose that λ < m,
λ 6= 0. If K = IBλ(L) in Rn, then K ∩ η = IBλ(L̃) in η, where the
body L̃ is defined by
(5.15) ρm−λ
(u) = c̃
Sn−1∩(η⊥⊕Ru)
ρn−λL (w)|u · w|
m−λ−1 dw,
u ∈ Sn−1 ∩ η, c̃ =
(m− λ) σn−m
2(n− λ)
if λ 6= −2ℓ, ℓ ∈ N,
π(m−n)/2 σn−m/2 otherwise.
Moreover, if K ∈ Inλ in Rn, then K ∩ η ∈ Imλ in η.
Proof. Let λ 6= −2ℓ, ℓ ∈ N, and let θ ∈ Sn−1 ∩ η. By Definition 5.7,
K = c
1−λρn−λL , and Theorem 3.12 (with f = sλρ
K and g =
c−1λ,nρ
L ) yields
K(θ) = (M
Sn−1∩η
T λη [c
L ])(θ) = c
λ,m(M
Sn−1∩η
)(θ),
where ρm−λ
= c T λη ρ
L , c = π
(n−m)/2(m − λ)/(n − λ). By Definition
5.7 and (3.20), we are done. If λ = −2ℓ, ℓ ∈ N, then, as above,
ρ−2ℓK (θ) = (M̃
Sn−1∩η
T−2ℓη ρ
L )(θ) = (M
Sn−1∩η
ρm+2ℓ
where ρm+2ℓ
= T−2ℓη ρ
L . This gives (5.15).
Furthermore, if K ∈ Inλ , λ 6= −2ℓ, ℓ ∈ N, then, by Definition 5.4,
K = M
1−λµ, µ ∈ Me+(Sn−1). Hence, by Theorem 3.12, there is
a measure ν ∈ Me+(Sn−1 ∩ η) such that the restriction of sλρλK onto
Sn−1∩η is represented as sλρλK =M1−λSn−1∩ην. It means that K∩η ∈ I
in η. In the case λ = −2ℓ, ℓ ∈ N, the argument is similar. �
6. Examples of λ-intersection bodies
The definition of the classes Inλ and IBλ,n and all known characteri-
zations are purely analytic. Unlike the case λ = 1, when an intersection
body of a star body is explicitly defined by a simple geometric proce-
dure, it is not clear how can we construct λ−intersection bodies in the
general case. Below we give some examples, when the radial function
of a λ−intersection body can be explicitly determined. These examples
utilize the generalized cosine transforms.
Example 6.1. Let λ < 1, λ 6= 0. This case is the simplest. Indeed,
given a non-negative measure µ on Sn−1, the relevant λ−intersection
28 BORIS RUBIN
body can be directly constructed by the formula ρλK = M
1−λµ, if λ 6=
−2ℓ, ℓ ∈ N, and ρ−2ℓK = M̃1+2ℓµ, otherwise. In other words (cf. (2.11)),
(6.1) ρλK(u) =
|θ · u|−λ dµ(θ).
This fact (with λ replaced by −p) is a reformulation of Theorem 6.17
from [K4], which was stated in the language of isometric embeddings
and relies on the P. Lévy characterization; see also Lemma 6.4 and
Theorem 4.11 in [K4].
Example 6.2. If n − 3 ≤ λ < n, λ > 0, then Inλ includes all origin-
symmetric convex bodies in Rn.
This fact is due to Koldobsky [K4, Corollary 4.9]. It can be proved
using a slight modification of the argument from [R3, Sec. 7] as follows.
By Theorem 5.1 (c), it suffices to check that for any o.s. convex body
K,M1+λ−nρλK ∈ Me+(Sn−1). For λ ≥ n−1, this is obvious. To handle
the case n− 3 ≤ λ < n− 1, suppose first that K is infinitely smooth.
Using polar coordinates, for Reα > 0, we can write
(6.2) (Mαρα+n−1K )(u) = (α + n− 1) γn(α)
|x · u|α−1 dx.
Then M1+λ−nρλK can be realized as analytic continuation (a.c.) at α =
1 + λ− n of the right-hand side of (6.2). The latter can be written as
I(α) = 2(α+ n− 1)γn(α)
tα−1AK,u(t) dt,
AK,u(t) = voln−1(K∩{tu+u⊥}). Taking analytic continuation (see [GS,
Chapter 1]), for −2 < α < 0 (which is equivalent to n−3 ≤ λ < n−1)
we get
a.c.I(α) = c1
tα−1[AK,u(t)−AK,u(0)] dt.
Similarly, a.c.I(α) at α = −2 (which corresponds to λ = n − 3) is
K,u(0). Following [GS], one can easily check that constants c1 and
c2 are negative. Since K is convex, both analytic continuations are
positive, and thereforeM1+λ−nρλK > 0. If K is an arbitrary o.s. convex
body, we approximate it in the radial metric by smooth o.s. convex
bodies Kj. Then for any test function ω ∈ D+(Sn−1), by the previous
step we have
(M1+λ−nρλK , ω) = (ρ
1+λ−nω) = lim
(ρλKj ,M
1+λ−nω)
= lim
(M1+λ−nρλKj , ω) ≥ 0.
INTERSECTION BODIES 29
Hence, by Theorem 9.1, M1+λ−nρλK is a non-negative measure and the
proof is complete.
Example 6.3. If ρλK =
Ri−λn−iν for some ν ∈ M+(Gn,n−i) and λ ≤ i < n,
then K ∈ Inλ .
Indeed, for any test function ω ∈ D(Sn−1), by (3.12) (with α = 1−λ)
we have
(ρλK , ω) = (
Ri−λn−iν, ω) = (ν, R
n−iω) = (ν
⊥, Ri−λn−i,⊥ω)
= c−1(ν⊥, RiM
1−λω) = c−1(R∗i ν
⊥,M1−λω), c =
2π(i−1)/2
It means that for 0 < λ ≤ i < n and ν ∈ M+(Gn,n−i),
(6.3) ρλK =
Ri−λn−iν ⇐⇒ {ρλK =M1−λµ, µ = c−1R∗i ν⊥}.
By Definition 5.4, this gives the result. The particular case λ = i
implies the embedding into Ini of the Zhang’s class Znn−i; see Definition
1.5. This embedding was proved in [K3] and [Mi1] in a different way;
see also [Mi2], where it is proved that Znn−i is a proper subset of Ini if
2 ≤ i ≤ n− 2.
Example 6.4. If 0 < (i− 1)/2 < λ ≤ i < n and ρλK = M i−λµ for some
µ ∈ M+(Sn−1), then K ∈ Inλ .
Indeed, by Lemma 3.4 (with α = i− λ, β = 1− λ), ρλK =M i−λµ =
M1−λAi−λ,1−λ, where Ai−λ,1−λ is an integral operator which preserves
positivity provided i− λ > 1− λ > 1− n, (i− λ) + (1− λ) < 2. This
is just our case.
Example 6.5. One can construct bodies K ∈ Inλ from bodies L ∈ Inδ
by the formula ρK = ρ
L provided 0 < δ < λ < n.
Indeed, by Definition 5.4, there is a measure µ ∈ M+(Sn−1) so that
ρδL = M
1−δµ. Then, by Lemma 3.4 (with α = 1 − δ, β = 1 − λ),
ρλK = ρ
L = M
1−δµ = M1−λA1−δ,1−λµ, and we are done. This example
generalizes the corresponding result from [Mi1, p. 533, Statement (c)],
which was obtained in a different way for the case, when λ and δ are
integers.
Example 6.6. Let
(6.4) Bnq = {x ∈ Rn : ||x||q =
|xk|q
≤ 1}.
If 0 < q ≤ 2, then Bnq ∈ Inλ for all λ ∈ (0, n). If 2 < q <∞, λ ∈ (0, n),
then Bnq ∈ Inλ if and only if λ ≥ n− 3.
30 BORIS RUBIN
Both statements are due to Koldobsky. The first one follows from
the fact that for 0 < q ≤ 2 the Fourier transform of ||x||−λq is a positive
S ′-distribution (see Lemmas 3.6 and 2.27 in [K4]). The second state-
ment is a reformulation of Theorem 4.13 from [K4]. The “if” part is a
consequence of Example 6.2.
7. (q, ℓ)-balls
In this section we consider one more example, which resembles Ex-
ample 6.6, but does not fall into its scope and requires a separate
consideration. Let
x = (x′, x′′) ∈ Rn, x′ ∈ Rn−ℓ =
Rej , x
′′ ∈ Rℓ =
j=n−ℓ+1
Rej ,
e1, . . . , εn being coordinate unit vectors. Consider the (q, ℓ)-ball
(7.1) Bnq,ℓ = {x : ||x||q,ℓ = (|x′|q + |x′′|q)1/q ≤ 1}, q > 0.
We wonder, for which triples (q, ℓ, n), Bnq,ℓ is a λ-intersection body. To
study this problem, we need some preparation. Consider the Fourier
integral
(7.2) γq,ℓ(η) =
e−|y|
eiy·η dy, η ∈ Rℓ, q > 0.
The function γq,ℓ(η) is uniformly continuous on R
ℓ and vanishes at
infinity.
Lemma 7.1. If 0 < q ≤ 2, then γq,ℓ(η) > 0 for all η ∈ Rℓ.
Proof. (Cf. [K4, p. 44, for ℓ = 1]). For η = 0, the statement is obvious.
It is known (see, e.g., [SW]), that
(7.3) [e−t|· |
]∧(η) = πℓ/2t−ℓ/2e−|η|
2/4t, t > 0.
This gives the result for q = 2. Let 0 < q < 2. By Bernstein’s theorem
[F, Chapter 18, Sec. 4], there is a non-negative finite measure µq on
[0,∞) so that e−zq/2 =
e−tz dµq(t), z ∈ [0,∞). Replace z by |y|2 to
(7.4) e−|y|
e−t|y|
dµq(t).
Then (7.3) yields
γq,ℓ(η) =
eiy·ηdy
e−t|y|
dµq(t) =
dµq(t)
eiy·ηe−t|y|
= πℓ/2
t−ℓ/2e−|η|
2/4t dµq(t) > 0.
INTERSECTION BODIES 31
The Fubini theorem is applicable here, because, by (7.4),
|eiy·η|dy
e−t|y|
dµq =
e−|y|
dy <∞.
Our next concern is the behavior of γq,ℓ(η) when |η| → ∞. If q
is even, then e−|·|
is a Schwartz function and therefore, γq,ℓ is infin-
itely smooth and rapidly decreasing. In the general case, we have the
following.
Lemma 7.2. For any q > 0,
(7.5) lim
|η|→∞
|η|ℓ+qγq,ℓ(η) = 2ℓ+qπℓ/2−1Γ(1+ q/2)Γ((ℓ+ q)/2) sin(πq/2).
Proof. For ℓ = 1, this statement can be found in [PS, Chapter 3, Prob-
lem 154] and in [K4, p. 45]. In the general case, the proof is more
sophisticated and relies on the properties of Bessel functions. By the
well-known formula for the Fourier transform of a radial function (see,
e.g., [SW]), we write γq,ℓ(η) = I(|η|), where
I(s) = (2π)ℓ/2s1−ℓ/2
rℓ/2Jℓ/2−1(rs) dr
= (2π)ℓ/2s−ℓ
[(rs)ℓ/2Jℓ/2(rs)] dr.
Integration by parts yields
I(s) = q(2π)ℓ/2s−ℓ/2
rℓ/2+q−1Jℓ/2(rs) dr.
Changing variable z = sqrq, we obtain
sℓ+qI(s) = (2π)ℓ/2A(s1/q), A(δ) =
e−zδzℓ/2qJℓ/2(z
1/q) dz.
We actually have to compute the limit A0 = lim
A(δ). To this end, we
invoke Hankel functions H
ν (z), so that Jν(z) = ReH
ν (z) if z is real
[Er]. Let hν(z) = z
ν (z). This is a single-valued analytic function
in the z-plane with cut (−∞, 0]. Using the properties of the Bessel
functions [Er], we get
(7.6) lim
hν(z) = 2
νΓ(ν)/πi,
(7.7) hν(z) ∼
2/π zν−1/2eiz−
(ν+ 1
), z → ∞.
Then we write A(δ) as A(δ) = Re
e−zδhℓ/2(z
1/q) dz and change the
line of integration from [0,∞) to ℓθ = {z : z = reiθ, r > 0} for
32 BORIS RUBIN
small θ < πq/2. By Cauchy’s theorem, owing to (7.6) and (7.7), we
obtain A(δ) = Re
e−zδhℓ/2(z
1/q) dz. Since for z = reiθ, hℓ/2(z
1/q) =
O(1) when r = |z| → 0 and hℓ/2(z1/q) = O(r(ℓ−1)/2qe−r
1/q sin(θ/q)) as
r → ∞, by the Lebesgue theorem on dominated convergence, we get
A0 = Re
hℓ/2(z
1/q) dz. To evaluate the last integral, we again use
analyticity and replace ℓθ by ℓπq/2 = {z : z = reiπq/2, r > 0} to get
A0 = Re
eiπq/2
hℓ/2(r
1/qeiπ/2) dr
To finalize calculations, we invoke McDonald’s function Kν(z) so that
hν(z) = z
νH(1)ν (z) = −
(ze−iπ/2)νKν(ze
−iπ/2).
This gives
sin(πq/2)
sℓ/2+q−1Kℓ/2(s) ds.
The last integral can be explicitly evaluated by the formula 2.16.2 (2)
from [PBM], and we obtain the result. �
Now we can proceed to studying (q, ℓ)-balls Bnq,ℓ; see (7.1). There is
an intimate connection between geometric properties of the balls Bnq,ℓ
and the Fourier transform of the power function || · ||pq,ℓ. The case q = 2
is well-known and associated with Riesz potentials; see, e.g., [St]. The
relevant case of ℓnq -balls, which agrees with ℓ = 1 was considered in
Example 6.6.
Lemma 7.3. Let q > 0, ξ = (ξ′, ξ′′) ∈ Rn, γq,ℓ(ξ′′) and γq,n−ℓ(ξ′) be
the functions of the form (7.2). We define
(7.8) hp,q,ℓ(ξ) =
Γ(−p/q)
tn+p−1 γq,n−ℓ(ξ
′t) γq,ℓ(ξ
′′t) dt.
(i) Let ξ′ 6= 0 and ξ′′ 6= 0. If q is even, then the integral (7.8) is abso-
lutely convergent for all p > −n. Otherwise, it is absolutely convergent
when −n < p < 2q. In these cases, hp,q,ℓ (ξ) is a locally integrable
function away from the coordinate subspaces Rℓ and Rn−ℓ.
(ii) If −n < p < 0, then hp,q,ℓ (ξ) ∈ L1loc(Rn)∩S ′(Rn) and (||·||
∧(ξ) =
hp,q,ℓ(ξ) in the sense of S ′-distributions. Specifically, for ϕ ∈ S(Rn),
(7.9) 〈hp,q,ℓ , ϕ̂〉 = (2π)n〈|| · ||pq,ℓ , ϕ〉.
INTERSECTION BODIES 33
Proof. (i) For any 0 < ε < a <∞,
ε<|ξ′|<a
ε<|ξ′′|<a
|hp,q,ℓ (ξ; , ξ′′)| dξ′′
|Γ(−p/q)|
tn+p−1 dt
ε<|ξ′|<a
|γq,n−ℓ (ξ′t)| dξ′
ε<|ξ′′|<a
|γq,ℓ (ξ′′t)| dξ′′
|Γ(−p/q)|
tp−1 dt
tε<|z′|<ta
|γq,n−ℓ (z′)| dz′
tε<|z′′|<ta
|γq,ℓ (z′′)| dz′′
|Γ(−p/q)|
(...) =
|Γ(−p/q)|
(I1 + I2).
The first integral is dominated by
tn+p−1 dt, c = σn−ℓ−1σℓ−1max
|γq,n−ℓ (z′)| max
|γq,ℓ (z′′)|
and is finite for p > −n. The second integral can be estimated by
making use of Lemma 7.2. Specifically, if q is not an even integer, then
I2 ≤ cε
tp−1 dt
|z′|>tε
|z′|n−ℓ+q
|z′′|>tε
|z′′|ℓ+q
tp−2q−1 dt.
If q is even, then γq,ℓ and γq,n−ℓ are rapidly decreasing and I2 ≤
tp−2m−1 dt for any m > 0. This gives what we need.
(ii) If −n < p < 0, the same argument is applicable with ε = 0. In
this case, I2 does not exceed ||γq,n−ℓ||1||γq,ℓ||1
tp−1 dt. The latter is
finite when p < 0, because, by Lemma 7.2, γq,n−ℓ and γq,ℓ are integrable
functions on respective spaces. When ξ → ∞, one can readily check
that hp,q,ℓ (ξ) = O(|ξ|m) for some m > 0, and therefore, hp,q,ℓ ∈ S ′(Rn).
To compute the Fourier transform (|| · ||pq,ℓ)∧(ξ), we replace ||x||
q,ℓ by
the formula
||x||pq,ℓ =
Γ(−p/q)
tp−1 e−|x
′/t|q−|x′′/t|q dt, p < 0,
34 BORIS RUBIN
and note that the Fourier transform of the function x→ e−|x′/t|q−|x′′/t|q
is just γq,n−ℓ (ξ
′t) γq,ℓ (ξ
′′t). Then
〈|| · ||pq,ℓ)
∧ , ϕ̂〉 = (2π)n〈|| · ||pq,ℓ , ϕ〉
(2π)nq
Γ(−p/q)
tp−1 dt
′/t|q−|x′′/t|qϕ(x) dx
Γ(−p/q)
tn+p−1 dt
γq,n−ℓ (ξ
′t) γq,ℓ (ξ
′′t) ϕ̂(ξ) dξ.
Interchange of the order of integration in this argument can be easily
justified using absolute convergence of integrals under consideration.
Theorem 7.4. If 0 < q ≤ 2, 0 < ℓ < n, then Bnq,ℓ is a λ-intersection
body for any 0 < λ < n.
Proof. Owing to Lemma 7.1, the function (7.8) (with p replaced by −λ)
is positive, and therefore, by Lemma 7.3, || · ||−λq,ℓ represents a positive
definite distribution. Now the result follows by Theorem 5.1. �
Consider the case q > 2. In this case Bnq,ℓ is convex, and, owing to
Example 6.2, Bnq,ℓ ∈ Inλ for all n− 3 ≤ λ < n. What about λ < n− 3?
This case is especially intriguing.
Proposition 7.5. If q > 2 and 0 < λ < max(n− ℓ, ℓ)− 2, then || · ||−λq,ℓ
is not a positive definite distribution and therefore, Bnq,ℓ 6∈ Inλ .
Proof. Let 0 < λ < n− ℓ− 2 and suppose the contrary, that Bnq,ℓ ∈ Inλ .
Consider the section of Bnq,ℓ by the (n − ℓ + 1)-dimensional plane η =
Ren ⊕ Rn−ℓ. By Theorem 5.12, Bnq,ℓ ∩ η ∈ In−ℓ+1λ in η, and therefore
||xnen + x′′||λq,ℓ = (|xn|q + |x′′|q)−λ/q
is a positive definite distribution in η. By the second derivative text (see
[K4, Theorem 4.19]) this is impossible if 0 < λ < n− ℓ− 2. A similar
contradiction can be obtained if we assume 0 < λ < ℓ− 2 and consider
the section of Bnq,ℓ by the (ℓ+ 1)-dimensional plane Re1 ⊕ Rℓ. �
Proposition 7.5 can be proved without using the second derivative
text and Theorem 5.12 on sections of λ-intersection bodies; see [R4].
The bounds for λ appear to be the same.
Open problem. Let q > 2, ℓ > 1. Is Bnq,ℓ a λ-intersection body if
max(n− ℓ, ℓ)− 2 < λ < n− 3?
This problem does not occur in the case ℓ = 1 as in Example 6.6.
INTERSECTION BODIES 35
8. The generalized cosine transforms and comparison of
volumes
For 1 < i < n, let voli(·) denote the i-dimensional volume function.
Suppose that i is fixed, and let A and B be o.s. convex bodies in Rn
satisfying
(8.1) voli(A ∩ ξ) ≤ voli(B ∩ ξ) ∀ξ ∈ Gn,i.
Does it follow that
(8.2) voln(A) ≤ voln(B) ?
This question is known as the Generalized Busemann-Petty Problem
(GBP); see [G], [RZ], [Z1].
Theorem 8.1. If GBP (8.1)-(8.2) has an affirmative answer, then
every smooth origin-symmetric convex body with positive curvature in
n is an (n− i)-intersection body.
Proof. Suppose that B is an o.s. convex body in Rn so that the radial
function ρB is infinitely smooth, the boundary of B has a positive curva-
ture and B /∈ Inn−i. By Definition 5.4, there is a function ϕ ∈ De(Sn−1),
which is negative on some open origin-symmetric set Ω ⊂ Sn−1 and
such that ρn−iB = M
1+i−nϕ. We choose a function h ∈ De(Sn−1) so
that h 6≡ 0, h(θ) ≥ 0 if θ ∈ Ω and h(θ) ≡ 0 otherwise. Define an o.s.
smooth body A by ρiA = ρ
B − εM1−ih, ε > 0. If ε is small enough,
then A is convex. Since by (3.12), RiM
1−ih = cR0n−i,⊥h ≥ 0, then
A ≤ RiρiB, which gives (8.1). On the other hand, by (3.5),
(ρn−iB , ρ
B − ρiA) = ε(M1+i−nϕ,M1−ih) = ε(ϕ, h) < 0,
or (ρn−iB , ρ
B) < (ρ
B , ρ
A). By Hölder’s inequality, this implies voln(B) <
voln(A), which contradicts (8.2). �
Remark 8.2. As we noted in Introduction, Theorem 8.1 is not new, and
its proof given in [K3] relies on a sequence of deep facts from functional
analysis. The proof presented above is much more elementary and
constructive. For instance, it allows us to keep invariance properties of
the bodies under control. This advantage was essentially used in our
paper [R4].
Theorem 8.1 and Proposition 7.5 imply the following
Corollary 8.3. Let 1 ≤ ℓ ≤ n/2; i > ℓ+2, B = Bn4,ℓ (see (7.1)). Then
there is a smooth o.s. convex body A in Rn so that (8.1) holds but (8.2)
fails.
36 BORIS RUBIN
Setting ℓ = 1 in this statement, we obtain the well-known Bourgain-
Zhang theorem, which states that GBP has a negative answer when
3 < i < n; see [BZ], [K4], [RZ] on this subject. For i = 2 and i = 3
(n ≥ 5) the GBP is still open. An affirmative answer in these cases
was obtained in [R4] for bodies having a certain additional symmetry.
9. Appendix
Every positive distribution F ∈ S ′(Rn) is given by a tempered non-
negative measure µ, i.e., 〈F, φ〉 =
φ(x)dµ(x); see, e.g., [GV, p.147]).
For convenience of the reader, we present a similar fact for the sphere.
Theorem 9.1. A distribution f ∈ D′(Sn−1) is positive if and only if
there is a measure µ ∈ M+(Sn−1) such that
(f, ϕ) =
ϕ(θ)dµ(θ) ∀ϕ ∈ D(Sn−1).
Proof. This statement is known, however, we could not find precise ref-
erence and decided to give a proof for convenience of the reader. The
“if” part is obvious. To prove the “ only if” part, we write a test func-
tion ϕ ∈ D(Sn−1) as a sum ϕ = ϕ1+iϕ2, where ϕ1 = Reϕ, ϕ2 = Imϕ.
Since −||ϕ||C(Sn−1) ≤ ϕj ≤ ||ϕ||C(Sn−1), j = 1, 2, and f is positive, then
−(f, 1) ||ϕ||C(Sn−1) ≤ (f, ϕj) ≤ (f, 1) ||ϕ||C(Sn−1),
and therefore, |(f, ϕ)| ≤ |(f, ϕ1)|+ |(f, ϕ2)| ≤ 2(f, 1) ||ϕ||C(Sn−1). Since
D(Sn−1) is dense in C(Sn−1), then f extends as a linear continuous
functional f̃ on C(Sn−1) and, by the Riesz theorem, there is a mea-
sure µ on Sn−1 such that (f̃ , ω) =
ω(θ)dµ(θ) for every ω ∈
C(Sn−1). In particular, (f, ϕ) = (f̃ , ϕ) =
ϕ(θ)dµ(θ) for every
ϕ ∈ D(Sn−1). By taking into account that every non-negative function
ω ∈ C(Sn−1) can be uniformly approximated by non-negative functions
ϕk ∈ D(Sn−1) (for instance, by Poisson integrals of ω), we get
ω(θ)dµ(θ) = lim
ϕk(θ)dµ(θ) = lim
(f, ϕk) ≥ 0.
The latter means that µ is non-negative. �
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Department of Mathematics, Louisiana State University, Baton Rouge,
LA, 70803 USA
E-mail address : borisr@math.lsu.edu

  Intersection bodies represent a remarkable class of geometric objects
associated with sections of star bodies and invoking
  Radon transforms, generalized cosine transforms, and the relevant Fourier
analysis. The main focus of this article is interrelation between generalized
cosine transforms of different kinds in the context of their application to
investigation of a certain family of intersection bodies, which we call
$\lam$-intersection bodies. The latter include $k$-intersection bodies (in the
sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of
$L_p$-spaces. In particular, we show that restrictions onto lower dimensional
subspaces of the spherical Radon transforms and the generalized cosine
transforms preserve their integral-geometric structure. We apply this result to
the study of sections of $\lam$-intersection bodies. New characterizations of
this class of bodies are obtained and examples are given. We also review some
known facts and give them new proofs.

Introduction.
2. Preliminaries.
3. Analytic families of the generalized cosine transforms.
4. Positive definite homogeneous distributions.
5. λ-intersection bodies.
6. Examples of λ-intersection bodies.
7. (q, ℓ)-balls.
8. The generalized cosine transforms and comparison of volumes.
9. Appendix.
2000 Mathematics Subject Classification. Primary 44A12; Secondary 52A38.
Key words and phrases. Spherical Radon transforms, cosine transforms, inter-
section bodies.
The research was supported in part by the NSF grant DMS-0556157 and the
Louisiana EPSCoR program, sponsored by NSF and the Board of Regents Support
Fund.
http://arxiv.org/abs/0704.0061v2
2 BORIS RUBIN
1. Introduction
This is an updated and extended version of our previous preprint
[R5].
Intersection bodies interact with Radon transforms and encompass
diverse classes of geometric objects associated to sections of star bodies.
The concept of intersection body was introduced in the remarkable
paper by Lutwak [Lu] and led to a breakthrough in the solution of the
long-standing Busemann-Petty problem; see [G], [K4], [Lu], [Z2] for
references and historical notes.
We remind some known facts that will be needed in the following.
An origin-symmetric (o.s.) star body in Rn, n ≥ 2, is a compact set
K with non-empty interior such that tK ⊂ K ∀t ∈ [0, 1], K = −K,
and the radial function ρK(θ) = sup{λ ≥ 0 : λθ ∈ K} is continuous
on the unit sphere Sn−1. In the following, Kn denotes the set of all
o.s. star bodies in Rn, Gn,i is the Grassmann manifold of i-dimensional
linear subspaces of Rn, and voli(·) denotes the i-dimensional volume
function. The Minkowski functional of a body K ∈ Kn is defined by
||x||K = min{a ≥ 0 : x ∈ aK}, so that ||θ||K = ρ−1K (θ), θ ∈ Sn−1.
Definition 1.1. [Lu] A body K ∈ Kn is an intersection body of a body
L ∈ Kn if ρK(θ) = voln−1(L ∩ θ⊥) for every θ ∈ Sn−1, where θ⊥ is the
central hyperplane orthogonal to θ.
By taking into account that voln−1(L ∩ θ⊥) in Definition 1.1 is a
constant multiple of the Minkowski-Funk transform
(Mf)(θ) =
Sn−1∩θ⊥
f(u) dθu, f(u) = ρ
L (u),
Goodey, Lutwak and Weil [GLW] generalized Definition 1.1 as follows.
Definition 1.2. A body K ∈ Kn is an intersection body if ρK = Mµ
for some even non-negative finite Borel measure µ on Sn−1.
A sequence of bodies Kj ∈ Kn is said to be convergent to K ∈ Kn
in the radial metric if lim
||ρKj − ρK ||C(Sn−1) = 0.
Proposition 1.3. The class of intersection bodies is the closure of the
class of intersection bodies of star bodies in the radial metric.
Proposition 1.4. If K is an intersection body in Rn, n > 2, then for
every i = 2, 3, . . . , n − 1 and every η ∈ Gn,i, K ∩ η is an intersection
body in η.
Regarding these two important propositions see [FGW], [GW] and
a nice historical survey in [G].
INTERSECTION BODIES 3
Different generalizations of the concept of intersection body associ-
ated to lower dimensional sections were suggested in the literature; see,
e.g., [K4], [RZ], [Z1]. The following one, which plays an important role
in the study of the lower dimensional Busemann-Petty problem, is due
to Zhang [Z1].
Definition 1.5. We say, that a body K ∈ Kn belongs to Zhang’s class
Zni if there is a non-negative finite Borel measure m on the Grassmann
manifold Gn,i such that ρ
K = R
im, where R
i is the dual spherical
Radon transform; see (2.2), (2.5).
Another generalization was suggested by Koldobsky [K2] and de-
scribed in detail in [K4]. This class of bodies will be our main concern.
Definition 1.6. [K4, p. 71] A body K ∈ Kn is a k-intersection body
of a body L ∈ Kn (we write K = IBk(L)) if
(1.1) volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) ∀ξ ∈ Gn,k.
We denote by IBk,n the set of all bodies K ∈ Kn satisfying (1.1) for
some L ∈ Kn.
When k = 1, this definition coincides with Definition 1.1 up to a
constant multiple. An analog of Definition 1.2 was given in the Fourier
analytic terms as follows.
Definition 1.7. [K4, Definition 4.7] A body K ∈ Kn is a k-intersection
body if there is a non-negative finite Borel measure µ on Sn−1, so that
for every Schwartz function φ,
||x||−kK φ(x) dx =
tk−1φ̂(tθ) dt
dµ(θ),
where φ̂ denotes the Fourier transform of φ.
The set of all k-intersection bodies in Rn will be denoted by Ink .
Keeping in mind Proposition 1.3 for k = 1, one can alternatively
define the class Ink as a closure of IBk,n in the radial metric; cf. [Mi1,
p. 532]. However, to apply results from [K4] to such class, equivalence
of this definition to Definition 1.7 must be proved. We will do this in
the more general situation in Section 5.2.
From Definitions 1.6 and 1.7 it is not clear, for which bodies L ∈ Kn
the relevant k-intersection body K = IBk(L) does exist. It is also not
obvious which bodies actually constitute the class Ink . The following
important characterization is due to Koldobsky.
4 BORIS RUBIN
Theorem 1.8. [K4, Theorem 4.8] A body K ∈ Kn is a k-intersection
body if and only if || · ||−kK represents a positive definite tempered dis-
tribution on Rn, that is, the Fourier transform (|| · ||−kK )∧ is a positive
tempered distribution on Rn.
The concept of k-intersection body is related to another important
development. For K ∈ Kn, the quasi-normed space (Rn, || · ||K) is said
to be isometrically embedded in Lp, p > 0, if there is a linear operator
T : Rn → Lp([0, 1]) so that ||x||K = ||Tx||Lp([0,1]).
Theorem 1.9. [K4, Theorem 6.10] The space (Rn, || · ||K) embeds iso-
metrically in Lp, p > 0, p 6= 2, 4, . . . , if and only if Γ(−p/2)(|| · ||pK)∧
is a positive distribution on Rn \ {0}.
Following Theorems 1.9 and 1.8, one can formally say that K ∈ Ink if
and only if (Rn, || · ||K) embeds isometrically in L−k. This observation,
combined with Definition 1.7, was used by A. Koldobsky to define the
concept of “isometric embedding in Lp” for negative p.
Definition 1.10. [K4, Definition 6.14] Let 0 < p < n, K ∈ Kn. The
space (Rn, || · ||K) is said to be isometrically embedded in L−p if there
is a non-negative finite Borel measure µ on Sn−1, so that for every
Schwartz function φ,
||x||−pK φ(x) dx =
tp−1φ̂(tθ) dt
dµ(θ),
where φ̂ denotes the Fourier transform of φ.
Origin-symmetric bodiesK in this definition can be regarded as “unit
balls of n-dimensional subspaces of L−p”. Comparing Definitions 1.10
and 1.7, one might call these bodies “p-intersection bodies”. Since the
meaning of the space L−p itself is not specified in Definition 1.10 and
since our paper is mostly focused on geometric properties of bodies
(rather than embeddings in Lp), in the following we prefer to adopt
another name “λ-intersection body”, where λ is a real number, that
will be specified in due course. We denote the set of all λ-intersection
bodies in Rn by Inλ .
Contents of the paper. We will focus on intimate connection
between intersection bodies, spherical Radon transforms, and general-
ized cosine transforms; see definitions in Section 2.2. This approach
is motivated by the fact that the volume of a central cross section of
a star body is expressed through the spherical Radon transform, and
the latter is a member of the analytic family of the generalized cosine
transforms. These transforms were introduced by Semyanistyi [Se] and
INTERSECTION BODIES 5
arise (up to naming and normalization) in different contexts of analysis
and geometry; see, e.g., [K4], [R1]-[RZ], [Sa2], [Sa3], [Str1], [Str2].
Sections 2-4 provide analytic background for geometric considera-
tions in Sections 5-7. In Section 2 we establish our notation and define
the generalized cosine transforms on the sphere and the relevant dual
transforms on Grassmann manifolds. In Section 3 we present basic
properties of these transforms, establish new relations between spheri-
cal Radon transforms and the generalized cosine transforms, and prove
“restriction theorems”, which are akin to trace theorems in Sobolev
spaces. Section 4 deals with positive definite homogeneous distribu-
tions, that can be characterized in terms of the generalized cosine
transforms. This section serves as a preparation for the forthcoming
definition of the concept of λ-intersection body. We investigate which
λ’s are appropriate and why. In Section 5 we switch to geometry and
define the class Inλ of λ-intersection bodies. The case 0 < λ < n cor-
responds to the “unit balls of L−p-spaces” in the spirit of Definition
1.10. The reader will find in this section new proofs of some known
facts. We introduce the notion of λ-intersection body of a star body
in Rn, which extends Definition 1.6 to all λ < n, λ 6= 0. The class
of all such bodies will be denoted by IBnλ . We will prove that for
all λ < n, λ 6= 0,−2,−4, . . . , the class Inλ is the closure of IBnλ in
the radial metric. The case λ = 1 gives Proposition 1.3. It will be
proved that all m-dimensional central sections of λ-intersection bod-
ies are λ-intersection bodies in the corresponding m-planes provided
λ < m, λ 6= 0.
The natural question arises: How to construct λ-intersection bodies?
In Section 6 we give a series of examples; some of them are known and
some are new. They can be obtained by utilizing auxiliary statements
from Section 3. In particular, the famous embedding of Zhang’s class
Znn−k into Ink , which was first established in [K3] and studied in [Mi1],
[Mi2], will be generalized to the case, when k is replaced by any λ ∈
(0, n). Section 7 is devoted to the so called (q, ℓ)-balls, defined by
Bnq,ℓ = {x = (x′, x′′) : |x′|q + |x′′|q ≤ 1; x′ ∈ Rn−ℓ, x′′ ∈ Rℓ}, q > 0.
We show that if 0 < q ≤ 2, then Bnq,ℓ ∈ Inλ for all λ ∈ (0, n). If
q > 2 and n − 3 ≤ λ < n, we still have Bnq,ℓ ∈ Inλ . If q > 2 and
0 < λ < λ0 = max(n − ℓ, ℓ) − 2, then Bnq,ℓ 6∈ Inλ . The case, when
q > 2, ℓ > 1, and λ0 ≤ λ < n− 3 represents an open problem.
In Section 8 we remind the generalized Busemann-Petty problem
(GBP) for i-dimensional central sections of o.s. convex bodies in Rn.
This challenging problem is still open for i = 2 and i = 3 (n ≥ 5).
It actually inspires the whole investigation. Using properties of the
6 BORIS RUBIN
generalized cosine transforms, we give a short direct proof of the fact
that an affirmative answer to GBP implies that every smooth o.s. con-
vex body in Rn with positive curvature is an (n− i)-intersection body.
This fact was discovered by A. Koldobsky. The original proof in [K3]
is based on the embedding Inn−i ⊂ Zni and Zhang’s result [Z1, Theorem
6]. The latter heavily relies on the Hahn-Banach separation theorem.
Our proof is more constructive and almost self-contained. We conclude
the paper by Appendix, which is added for convenience of the reader.
The list of references at the end of the paper is far from being com-
plete. Further references can be found in cited books and papers.
Acknowledgement. I am grateful to Professor Alexander Koldob-
sky, who shared with me his knowledge of the subject. Special thanks
go to Professors Erwin Lutwak, Deane Yang, and Gaoyong Zhang for
useful discussions.
2. Preliminaries
2.1. Notation. In the following, N = {1, 2, . . . } is the set of all nat-
ural numbers, Sn−1 is the unit sphere in Rn with the area σn−1 =
2πn/2/Γ(n/2); Ce(S
n−1) is the space of even continuous functions on
Sn−1; SO(n) is the special orthogonal group of Rn; for θ ∈ Sn−1 and
γ ∈ SO(n), dθ and dγ denote the relevant invariant probability mea-
sures; D(Sn−1) is the space of C∞-functions on Sn−1 equipped with
the standard topology, and D′(Sn−1) stands for the corresponding dual
space of distributions. The subspaces of even test functions (distribu-
tions) are denoted by De(Sn−1) ( D′e(Sn−1)); Gn,i denotes the Grass-
mann manifold of i-dimensional subspaces ξ of Rn with the SO(n)-
invariant probability measure dξ; D(Gn,i) is the space of infinitely dif-
ferentiable functions on Gn,i.
We write M(Sn−1) and M(Gn,i) for the spaces of finite Borel mea-
sures on Sn−1 and Gn,i; M+(Sn−1) and M+(Gn,i) are the relevant
spaces of non-negative measures; Me+(Sn−1) denotes the space of even
measures µ ∈ M+(Sn−1). Given a function ϕ on Gn,i, we denote
ϕ⊥(η) = ϕ(η⊥), η ∈ Gn,n−i. Similarly, given a measure µ ∈ M(Gn,n−i),
the corresponding “orthogonal measure” µ⊥ in M(Gn,i) is defined by
(µ⊥, ϕ) = (µ, ϕ⊥), ϕ ∈ C(Gn,i).
Let {Yj,k} be an orthonormal basis of spherical harmonics on Sn−1.
Here j = 0, 1, 2, . . . , and k = 1, 2, . . . , dn(j), where dn(j) is the di-
mension of the subspace of spherical harmonics of degree j. Each
function ω ∈ D(Sn−1) admits a decomposition ω =
j,k ωj,kYj,k with
the Fourier-Laplace coefficients ωj,k =
ω(θ)Yj,k(θ)dθ, which decay
rapidly as j → ∞. Each distribution f ∈ D′(Sn−1) can be defined by
INTERSECTION BODIES 7
(f, ω) =
j,k fj,kωj,k where fj,k = (f, Yj,k) grow not faster than j
m for
some integer m. We will need the Poisson integral, which is defined for
f ∈ L1(Sn−1) by
(2.1) (Πtf)(θ) = (1− t2)
f(u)|θ − tu|−ndu, 0 < t < 1,
and has the Fourier-Laplace decomposition Πtf =
j,k t
jfj,kYj,k [SW].
For f ∈ D′(Sn−1), this decomposition serves as a definition of Πtf . For
harmonic analysis on the unit sphere, the reader is referred to [Le],
[Mü], [Ne], [SW], and a survey article [Sa3].
2.2. Basic integral transforms. For integrable functions f on Sn−1
and ϕ onGn,i, 1 ≤ i ≤ n−1, the spherical Radon transform (Rif)(ξ), ξ ∈
Gn,i, and its dual (R
iϕ)(θ), θ ∈ Sn−1, are defined by
(2.2) (Rif)(ξ) =
θ∈Sn−1∩ξ
f(θ) dξθ, (R
iϕ)(θ) =
ϕ(ξ) dθξ,
where dξθ and dθξ denote the probability measures on the manifolds
Sn−1 ∩ ξ and {ξ ∈ Gn,i : ξ ∋ θ}, respectively. The precise meaning of
the second integral is
(2.3) (R∗iϕ)(θ) =
SO(n−1)
ϕ(rθγp0) dγ, θ ∈ Sn−1,
where p0 is an arbitrarily fixed coordinate i-plane containing the north
pole en and rθ ∈ SO(n) is a rotation satisfying rθen = θ.
Operators Ri and R
i extend to finite Borel measures in a canonical
way, using the duality
(2.4)
(Rif)(ξ)ϕ(ξ)dξ =
f(θ)(R∗iϕ)(θ)dθ.
Specifically, for µ ∈ M(Sn−1) and m ∈ M(Gn,i), we define Riµ ∈
M(Gn,i) and R∗im ∈ M(Sn−1) by
(2.5) (Riµ, ϕ)=
(R∗iϕ)(θ)dµ(θ), (R
im, f)=
(Rif)(ξ)dm(ξ),
where ϕ ∈ C(Gn,i), f ∈ C(Sn−1).
The generalized cosine transforms are defined by
(2.6) (Rαi f)(ξ) = γn,i(α)
|Prξ⊥θ|α+i−n f(θ) dθ,
(2.7) (
αϕ)(θ) = γn,i(α)
|Prξ⊥θ|α+i−n ϕ(ξ) dξ,
8 BORIS RUBIN
γn,i(α) =
σn−1 Γ((n− α− i)/2)
2π(n−1)/2 Γ(α/2)
, Re α > 0, α+i−n 6= 0, 2, 4, . . . .
Here Prξ⊥θ stands for the orthogonal projection of θ onto ξ
⊥, the or-
thogonal complement of ξ ∈ Gn,i. If f and ϕ are smooth enough, then
integrals (2.2) can be regarded (up to a constant multiple) as members
of the relevant analytic families (2.6) and (2.7); cf. Lemma 3.1. The
particular case i = n − 1 in (2.2) corresponds to the Minkowski-Funk
transform
(2.8) (Mf)(u) =
{θ : θ·u=0}
f(θ) duθ = (Rn−1f)(u
⊥), u ∈ Sn−1,
which integrates a function f over great circles of codimension 1. This
transform is a member of the analytic family
(2.9) (Mαf)(u) = (Rαn−1f)(u
⊥) = γn(α)
f(θ)|θ · u|α−1 dθ,
(2.10) γn(α)=
σn−1 Γ
(1−α)/2
2π(n−1)/2Γ(α/2)
, Re α>0, α 6=1, 3, 5, . . . .
The values α = 1, 3, 5, . . . are poles of the Gamma function Γ((1−α)/2).
In some occasions we include these values into consideration and set
(2.11) (M̃αf)(u) =
f(θ)|θ · u|α−1 dθ.
Historical notes. Regarding spherical Radon transforms (2.2) and
the Minkowski-Funk transform (2.8), see [GGG], [He], [R2], [R3]. The
first detailed investigation of the analytic family {Mα} is due to Se-
myanistyi [Se], who showed that these operators naturally arise in the
Fourier analysis of homogeneous functions. The case α = 2 in (2.11)
was known before, thanks to W. Blaschke, A.D. Alexandrov, and P.
Lévy. Integrals (2.9) (sometimes with different normalization) arise
in diverse areas of analysis and geometry; see [K4], [R1] - [R3], [Sa3],
[Str1], and references therein. In convex geometry and Banach space
theory, operators (2.11) with α − 1 replaced by p are known as the p-
cosine transforms. More general analytic families (2.6) and (2.7) were
introduced in [R2].
INTERSECTION BODIES 9
3. Analytic Families of the Generalized Cosine
Transforms
3.1. Basic properties. Below we review basic properties of integrals
(2.6), (2.7), (2.9); see [R2], [R3] for more details. For integrable func-
tions f and ϕ and Reα > 0, integrals (2.6), (2.7) and (2.9) are ab-
solutely convergent. When f and ϕ are infinitely differentiable, these
integrals extend meromorphically to all α ∈ C.
Lemma 3.1. If f and ϕ are continuous functions, then
Rαi f = R
i f = ciRif, ci =
2π(i−1)/2
;(3.1)
0ϕ = ciR
iϕ,(3.2)
Mαf = M0f = cn−1Mf, cn−1 =
2π(n−2)/2
.(3.3)
Hence, the Radon transform, its dual, and the Minkowski-Funk trans-
form can be regarded (up to a constant multiple) as members of the
corresponding analytic families {Rαi }, {
α}, {Mα}.
Proof. Formulas (3.2) and (3.3) follow from (3.1). To prove (3.1), we
write (2.6) in bi-spherical coordinates θ = u sin ψ + vcosψ, where
u ∈ Sn−1 ∩ ξ ∼ Si−1, v ∈ Sn−1 ∩ ξ⊥ ∼ Sn−i−1, 0 ≤ ψ ≤ π/2.
dθ = c sini−1 ψ cosn−i−1ψ dψdudv, c = σi−1σn−i−1/σn−1.
This gives
(Rαi f)(ξ) = c γn,i(α)
∫ π/2
sini−1 ψ cosα−1ψ dψ
Sn−1∩ξ⊥
Sn−1∩ξ
f(u sin ψ+vcosψ) du
ci(α)
Γ(α/2)
tα/2−1F (t) dt,
where
ci(α) =
c γn,i(α) Γ(α/2)
σi−1σn−i−1
Γ((n− α− i)/2)
2π(n−1)/2
2π(i−1)/2
as α → 0, and
F (t) = (1− t2)i/2−1
Sn−1∩ξ⊥
Sn−1∩ξ
1− t2+vt) du.
10 BORIS RUBIN
Since
Γ(α/2)
tα/2−1F (t) dt = F (0) =
Sn−1∩ξ
f(u)du = (Rif)(ξ),
we are done. �
Analytic continuation of integrals (2.9) can be realized in spherical
harmonics as Mαf=
mj,αfj,kYj,k, where
(3.4) mj,α=
(−1)j/2
Γ(j/2 + (1− α)/2)
Γ(j/2 + (n− 1 + α)/2)
if j is even,
0 if j is odd;
see [R1], [R3]. If f ∈D′(Sn−1), then Mαf is a distribution defined by
(Mαf, ω)=(f,Mαω)=
mj,α fj,k ωj,k, ω∈D(Sn−1); α 6=1, 3, 5, . . . .
Lemma 3.2. Let α, β ∈ C; α, β 6= 1, 3, 5, . . . . If α + β = 2 − n and
f ∈ De(Sn−1) (or f ∈ D′e(Sn−1)), then
(3.5) MαMβf = f.
If α, 2−n−α 6= 1, 3, 5, . . ., then Mα is an automorphism of the spaces
De(Sn−1) and D′e(Sn−1).
Proof. The equality (3.5) is equivalent to mj,αmj,β = 1, α+β = 2−n.
The latter follows from (3.4). The second statement is a consequence of
the standard theory of spherical harmonics [Ne], because the Fourier-
Laplace multiplier mj,α has a power behavior as j → ∞. �
Corollary 3.3. The Minkowski-Funk transform on the spaces De(Sn−1)
and D′e(Sn−1) can be inverted by the formula
(3.6) (M)−1 = cn−1M
2−n, cn−1 =
2π(n−2)/2
Note that there is a wide variety of diverse inversion formulas for
the Minkowski-Funk transform (see [GGG], [He], [R3] and references
therein), but all of them are, in fact, different realizations of (3.6),
depending on classes of functions.
3.2. Auxiliary statements. We establish some connections between
operator families defined above.
INTERSECTION BODIES 11
Lemma 3.4. Let α, β ∈ C; α, β 6= 1, 3, 5, . . . . If Reα > Reβ, then
Mα =MβAα,β, where Aα,β is a spherical convolution operator with the
Fourier-Laplace multiplier
(3.7) aα,β(j) =
Γ(j/2 + (1− α)/2)
Γ(j/2 + (n− 1 + α)/2)
Γ(j/2 + (n− 1 + β)/2)
Γ(j/2 + (1− β)/2)
so that aα,β(j) ∼ (j/2)β−α as j → ∞. If α and β are real numbers
satisfying α > β > 1− n, α + β < 2, then Aα,β is an integral operator
such that Aα,βf ≥ 0 for every non-negative f ∈ L1(Sn−1).
Proof. The first statement follows from (3.4). To prove the second one,
we consider integral operators
+ f)(x) =
Γ(µ/2)
(1− t2)µ/2−1(Πtf)(x) tn−νdt,(3.8)
− f)(x) =
Γ(µ/2)
(t2 − 1)µ/2−1(Π1/tf)(x) t1−νdt,(3.9)
expressed through the Poisson integral (2.1). The Fourier-Laplace mul-
tipliers of Q
+ and Q
− are
(3.10) q̂
+ (j)=
Γ((j+n−ν+1)/2)
Γ((j+n−ν+1+µ)/2)
− (j)=
Γ((j+ν−µ)/2)
Γ((j+ν)/2)
They can be easily computed by taking into account that Πt ∼ tj in
the Fourier-Laplace terms. If f ∈ L1(Sn−1) and 0 < µ < ν < n, then
integrals (3.8) and (3.9) are absolutely convergent and obey Q
± f ≥ 0
when f ≥ 0. Comparing (3.10) and (3.7), we obtain a factorization
Aα,β = Q
α−β,1−β
α−β,1−β
− (set µ = α − β, ν = 1 − β), which implies
the second statement of the lemma. �
It is convenient to introduce a special notation for the spherical
Radon transform and the generalized cosine transform with orthogonal
argument. Assuming ξ ∈ Gn,i, we denote
(3.11) (Rn−i,⊥f)(ξ) = (Rn−if)(ξ
⊥), (Rαn−i,⊥f)(ξ) = (R
n−if)(ξ
Lemma 3.5. Let f ∈ L1(Sn−1), Re α > 0; α 6= 1, 3, 5, . . . . Then
(3.12) (RiM
αf)(ξ) = c (Rα+i−1n−i,⊥ f)(ξ), ξ ∈ Gn,i, c =
2π(i−1)/2
or (replace i by n− i)
(3.13) (Rn−i,⊥M
αf)(ξ) =
2π(n−i−1)/2
σn−i−1
(Rα+n−i−1i f)(ξ).
If f ∈ De(Sn−1), then (3.12) and (3.13) extend to Reα ≤ 0 by analytic
continuation.
12 BORIS RUBIN
Proof. For Reα > 0,
αf)(ξ) = γn(α)
Sn−1∩ξ
f(θ)|θ · u|α−1 dθ.
Since |θ · u| = |Prξθ||vθ · u| for some vθ ∈ Sn−1 ∩ ξ, by changing the
order of integration, we obtain
αf)(ξ) = γn(α)
f(θ)|Prξθ|α−1 dθ
Sn−1∩ξ
|vθ · u|α−1dξu.
The inner integral is independent on vθ and can be easily evaluated:
Sn−1∩ξ
|vθ · u|α−1dξu =
|t|α−1(1− t2)(i−3)/2 dt
2π(i−1)/2 Γ(α/2)
σi−1 Γ((i+ α− 1)/2)
This implies (3.12). �
The following statement is dual to Lemma 3.5.
Lemma 3.6. Let µ ∈ M(Gn,i), α 6= 1, 3, 5, . . . . Then
(3.14) MαR∗iµ = c
Rα+i−1n−i µ
⊥, c = 2π(i−1)/2/σi−1,
in the D′(Sn−1)-sense. If Reα > 0 and µ is absolutely continuous with
density ϕ ∈ L1(Gn,i), then
(3.15) MαR∗iϕ = c
Rα+i−1n−i ϕ
almost everywhere on Sn−1. If ϕ ∈ D(Gn,i), then (3.15) extends to all
complex α 6= 1, 3, 5, . . . by analytic continuation.
Proof. Let ω ∈ De(Sn−1) (it suffices to consider only even test func-
tions). By (2.4) and (3.12),
(MαR∗iµ, ω) = (µ,RiM
αω) = c (µ,Rα+i−1n−i,⊥ ω) = c (µ
⊥, Rα+i−1n−i ω).
This gives the result. �
The next statement contains explicit representations of the right in-
verse of the dual Radon transform R∗i (note that R
i is non-injective on
D(Gn,i) when 1 < i < n− 1).
Lemma 3.7. Every function f ∈De(Sn−1) is represented as f=R∗iAf ,
where A : De(Sn−1) → D(Gn,i),
(3.16) Af = c1R
i f = c2Rn−i,⊥M
2−nf,
π(1−i)/2σn−2
σn−i−1
Γ((n− i)/2)
Γ((n− 1)/2)
, c2 =
2πn/2−1
INTERSECTION BODIES 13
Proof. The coincidence of expressions in (3.16) follows from (3.13). To
prove the first equality, we invoke spherical convolutions defined by
analytic continuation of the integral
(3.17) (Qαf)(θ)=
σn−1Γ((n−1−α)/2)
2π(n−1)/2Γ(α/2)
(1−|u·θ|2)(α−n+1)/2f(u)du,
Reα > 0, α − n 6= 0, 2, 4, . . . , so that Q0f = f [R2]. By Theorem
1.1 from [R2], R∗iR
i f = c
α+i−1f , and therefore (set α = 1 − i),
i f = c
1 f , as desired. �
The next statement provides an intriguing factorization of the Minkowski-
Funk transform in terms of Radon transforms associated to mutually
orthogonal subspaces. This factorization can be useful in different oc-
currences.
Theorem 3.8. For f ∈ L1(Sn−1) and 0 < i < n,
(3.18) Mf = R∗iRn−i,⊥f.
Proof. By (2.3),
(R∗iRn−i,⊥f)(θ) =
SO(n−1)
(Rn−i,⊥f)(rθγR
i) dγ
SO(n−1)
(Rn−if)(rθγR
n−i) dγ
SO(n−1)
Sn−1∩rθγR
f(v) dv
Sn−1∩Rn−i
SO(n−1)
f(rθγw) dγ.
The inner integral is independent on w ∈ Sn−1 ∩ Rn−i and equals
(Mf)(θ). This gives (3.18). �
3.3. Restriction theorems. Theorems of such type deal with traces
of functions on lower dimensional subspaces and are well known, for in-
stance, in the theory of function spaces. To the best of our knowledge,
traces of functions represented by Radon transforms or, more generally,
by the generalized cosine transforms , were not studied systematically
and deserve particular attention, because they provide analytic back-
ground to a series of results related to sections of star bodies; cf. [R3,
Sec. 3.5], [FGW]. Given a subspace η ∈ Gn,m and k < m, we denote
by Gk(η) the manifold of all k-dimensional subspaces of η.
14 BORIS RUBIN
Theorem 3.9. Let f ∈ Ce(Sn−1), 1 ≤ k < m < n, λ 6= 0,−2,−4, . . . .
If Reλ < k, then for every η ∈ Gn,m and every ξ ∈ Gk(η),
(3.19) (Rk−λn−kf)(ξ
⊥) = (Rk−λm−kT
η f)(ξ
⊥ ∩ η),
where
(3.20) (T λη f)(u) = c̃
Sn−1∩(η⊥⊕Ru)
f(w)|u · w|m−λ−1 dw,
u ∈ Sn−1 ∩ η, c̃ = π(m−n)/2 σn−m/2.
In particular (let λ→ k),
(3.21) (Rn−kf)(ξ
⊥)=c (Rm−kT
η f)(ξ
⊥ ∩ η), c= π
(n−m)/2 σm−k−1
σn−k−1
Proof. By (2.6),
(Rk−λn−kf)(ξ
⊥)=γn,n−k(k − λ)
|Prξθ|−λ f(θ) dθ.
We represent θ in bi-spherical coordinates as
(3.22) θ = ucosψ + v sinψ,
where
u ∈ Sn−1 ∩ η ∼ Sm−1, v ∈ Sn−1 ∩ η⊥ ∼ Sn−m−1, 0 ≤ ψ ≤ π/2,
dθ = c′′ sinn−m−1 ψ cosm−1ψ dψdudv, c′′ = σm−1σn−m−1/σn−1.
If ξ ⊂ η, then |Prξθ| = |Prξ[Prηθ]| = |Prξu| cosψ, and therefore,
(Rk−λn−kf)(ξ
⊥) = γm,m−k(k − λ)
Sn−1∩η
|Prξu|−λ(T λη f)(u) du,
where
(T λη f)(u) =
c′′ γn,n−k(k − λ)
γm,m−k(k − λ)
∫ π/2
sinn−m−1 ψ cosm−λ−1ψ dψ
Sn−1∩η⊥
f(ucosψ+v sin ψ) dv
π(m−n)/2 σn−m
Sn−1∩(η⊥⊕Ru)
f(w)|u · w|m−λ−1 dw.
Formula (3.21) follows from (3.19) by (3.1). �
INTERSECTION BODIES 15
Theorem 3.10. Let f ∈De(Sn−1), η∈Gn,m, 1<m<n. Suppose that
f =M1−λg, where Reλ < m, λ 6= 0,−2,−4, . . . . Then the restriction
of f onto η is represented as f =M1−λ
Sn−1∩η
T λη g, where T
η has the form
(3.20) andM1−λ
Sn−1∩η
denotes the same operator M1−λ, but on the sphere
Sn−1 ∩ η.
Proof. For Reλ < 1, the statement is a particular case of Theorem
3.9 (set k = 1). For other values of λ, the result follows by analytic
continuation. �
Remark 3.11. The restriction λ 6= 0,−2,−4, . . . in Theorems 3.9 and
3.10 is caused by the Gamma function Γ(λ/2) in the numerator of the
corresponding normalizing factor. It is evident from the proof, that
both theorems remain true also for λ = −2ℓ, ℓ ∈ N, if we remove the
normalizing factor. Then M1−λ in Theorem 3.10 will be substituted
for M̃1+2ℓ; see (2.11).
We will need the following generalization of Theorem 3.10.
Theorem 3.12. Let f ∈Ce(Sn−1), µ ∈ Me+(Sn−1), and let η∈Gn,m,
1<m<n. Suppose that f =M1−λµ, if λ < m, λ 6= −2ℓ, ℓ ∈ N, and
f=M̃1+2ℓµ, if λ = −2ℓ.
(i) There is a measure ν ∈ Me+(Sn−1 ∩ η) such that the restriction of
f onto Sn−1 ∩ η is represented as f =M1−λ
Sn−1∩η
(ii) If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then (i) holds with dν(θ) =
(T λη g)(θ)dθ, where T
η g has the form (3.20).
(iii) If λ = −2ℓ, ℓ ∈ N, then (i) and (ii) hold with M1−λ
Sn−1∩η
substituted
for M̃1+2ℓ
Sn−1∩η
Proof. STEP 1. Let first λ < m, λ 6= 0,−2,−4, . . . . We invoke the
Poisson integral (2.1) so that
Πtf = ΠtM
1−λµ =M1−λgt, gt = Πtµ ∈ De(Sn−1), t ∈ (0, 1).
Since f is continuous, then Πtf converges to f as t → 0 uniformly
on Sn−1, and therefore, uniformly on Sn−1 ∩ η. Hence, for any test
function ω ∈ D(Sn−1 ∩ η), owing to Theorem 3.10, we have
(f, ω) = lim
(Πtf, ω) = lim
(M1−λgt, ω)
= lim
(M1−λ
Sn−1∩η
T λη gt, ω) = lim
(T λη gt,M
Sn−1∩η
= lim
(νt,M
Sn−1∩η
ω), νt = T
η gt.(3.23)
Thus, lim
(νt,M
Sn−1∩η
ω) exists for every ω∈D(Sn−1∩η). If ω is even,
i.e., ω ∈ De(Sn−1 ∩ η), then, by Lemma 3.2, we can replace ω by
16 BORIS RUBIN
M1−m+λ
Sn−1∩η
ω and conclude that the limit lim
(νt, ω) is well-defined for every
ω ∈ De(Sn−1∩η). Since νt = T λη Πtµ is an even function and the generic
test function ω ∈ D(Sn−1 ∩ η) can be represented as ω+ + ω−, where
ω± are even and odd, respectively, it follows that the limit lim
(νt, ω) =
(νt, ω+) is well-defined for every ω ∈ D(Sn−1∩ η) (not only for even
ω, as stated above). Since D′(Sn−1∩ η) is weakly complete, there is an
even distribution ν in D′(Sn−1 ∩ η) so that
(ν, ω) = lim
(νt, ω), ω ∈ D(Sn−1 ∩ η).
Furthermore, since (νt, ω) = (T
η Πtµ, ω) is non-negative for every non-
negative ω ∈ D(Sn−1 ∩ η) and every t ∈ (0, 1), then ν is a positive
distribution and, by Theorem 9.1, ν is a measure in Me+(Sn−1 ∩ η).
Thus, by (3.23), (f, ω) = lim
(νt,M
Sn−1∩η
ω) = (ν,M1−λ
Sn−1∩η
ω), which
means that f =M1−λ
Sn−1∩η
ν, as desired.
If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then νt = T λη Πtg tends to T λη g uni-
formly on Sn−1∩η as t→ 0. Hence, by (3.23), (f, ω) = (T λη g,M1−λSn−1∩ηω),
which means f =M1−λ
Sn−1∩η
T λη g.
STEP 2. Consider the case λ = −2ℓ, ℓ ∈ N, when f = M̃1+2ℓµ,
µ ∈ Me+(Sn−1), and the operator T λη = T−2ℓη has the form
(T−2ℓη h)(u) = c̃
Sn−1∩(η⊥⊕Ru)
|u · w|m+2ℓ−1 h(w) dw,
cf. (3.20). For any functions h ∈ C(Sn−1) and ω ∈ C(Sn−1 ∩ η),
(3.24) (T−2ℓη h, ω) = (h,
T−2ℓη ω),
where
T−2ℓη ω)(θ)=
Γ(m/2)
2Γ(n/2)
( Prηθ
|Prηθ|
|Prηθ|2ℓ ∈ C(Sn−1).
INTERSECTION BODIES 17
Indeed, using bi-spherical coordinates (see (3.22)), we have
(T−2ℓη h, ω) = c̃
Sn−1∩η
ω(u)du
Sn−1∩(η⊥⊕Ru)
h(w)|u · w|m+2ℓ−1 dw
c̃ σn−m−1
Sn−1∩η
ω(u)du
sinn−m−1 ψ cosm+2ℓ−1ψ dψ
Sn−1∩η⊥
h(ucosψ+v sin ψ) dv
c̃ σn−m−1
c′′ σn−m
h(θ)ω
( Prηθ
|Prηθ|
|Prηθ|2ℓ dθ = (h,
T−2ℓη ω).
Let h = Πtµ and observe that the limit lim
(T−2ℓη Πtµ, ω) exists, be-
cause, by (3.24), (T−2ℓη Πtµ, ω) = (Πtµ,
T−2ℓη ω) → (µ,
T−2ℓη ω). Note
that (T−2ℓη Πtµ, ω) ≥ 0 for any non-negative ω ∈ C(Sn−1∩η). Applying
the standard completeness argument (as in Step 1), we conclude, that
there is a measure ν ∈ M+(Sn−1 ∩ η) such that
(T−2ℓη Πtµ, ω) = (ν, ω) ∀ω ∈ C(Sn−1 ∩ η).
Using this equality, for f = M̃1+2ℓµ we obtain
(f, ω) = lim
(Πtf, ω) = lim
(ΠtM̃
1+2ℓµ, ω) = lim
(M̃1+2ℓΠtµ, ω)
(use Theorem 3.10 and Remark 3.11)
= lim
(M̃1+2ℓ
Sn−1∩η
T−2ℓη Πtµ, ω) = lim
(T−2ℓη Πtµ, M̃
Sn−1∩η
= (ν, M̃1+2ℓ
Sn−1∩η
This gives the result.
If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then, by Theorem 3.10 and Remark
3.11, for θ ∈ Sn−1 ∩ η we have
(Πtf)(θ) = (ΠtM̃
1+2ℓg)(θ) = (M̃1+2ℓΠtg)(θ) = (M̃
Sn−1∩η
T−2ℓη Πtg)(θ).
Owing to continuity of the operators M̃1+2ℓ
Sn−1∩η
, T−2ℓη , and Πt in the
relevant spaces of continuous functions, by passing to the limit as t→ 0,
we obtain f(θ) = (M̃1+2ℓ
Sn−1∩η
T−2ℓη g)(θ), θ ∈ Sn−1 ∩ η, as desired. �
18 BORIS RUBIN
4. Positive Definite Homogeneous Distributions
We remind some known facts; see, e.g., [GS], [Le]. Let S(Rn) be the
Schwartz space of rapidly decreasing C∞-functions on Rn and S ′(Rn)
its dual. The Fourier transform of F ∈ S ′(Rn) is defined by
〈F̂ , φ̂〉 = (2π)n〈F, φ〉, φ̂(y) =
φ(x) eix·y dx, φ ∈ S(Rn).
A distribution F ∈ S ′(Rn) is homogeneous of degree λ ∈ C if for any
φ ∈ S(Rn) and any a > 0, 〈F, φ(x/a)〉 = aλ+n 〈F, φ〉. Homogeneous dis-
tributions on Rn are intimately connected with distributions on Sn−1.
Let first f ∈ L1(Sn−1), (Eλf)(x) = |x|λf(x/|x|), x ∈ Rn \ {0}. The
operator Eλ generates a meromorphic S ′-distribution
〈Eλf, φ〉= a.c.
rλ+n−1u(r)dr, u(r) =
f(θ)φ(rθ)dθ,
where “a.c.” denotes analytic continuation in the λ-variable. The dis-
tribution Eλf is regular if Reλ > −n and admits simple poles at
λ = −n,−n − 1, . . .. The above definition extends to all distributions
f ∈ D′(Sn−1) by the formula
〈Eλf, φ〉 = a.c.
rλ+n−1u(r)dr, u(r) = (f, φ(rθ)), 1
and the map Eλ : D′(Sn−1) → S ′(Rn) is weakly continuous. If f
is orthogonal to all spherical harmonics of degree j, then the deriv-
ative u(j)(r) equals zero at r = 0 and the pole at λ = −n − j is
removable. In particular, if f is an even distribution, i.e., (f, ϕ) =
(f, ϕ−), ϕ−(θ) = ϕ(−θ) ∀ϕ ∈ D(Sn−1), then the only possible poles
of Eλf are −n,−n− 2,−n− 4, . . . .
The Fourier transform of homogeneous distributions was extensively
studied by many authors; see [Sa3] and references therein. We restrict
our consideration to even distributions, when the operator family {Mα}
defined by (2.9) naturally arises thanks to the formula
(4.1) [E1−n−αf ]
∧ = 21−απn/2Eα−1M
This formula amounts to Semyanistyi [Se]. If f ∈ De(Sn−1), then (4.1)
holds pointwise for 0 < Reα < 1 (see, e.g., Lemma 3.3 in [R1] ) and
extends in the S ′-sense to all α ∈ C satisfying
(4.2) α /∈ {1, 3, 5, . . .} ∪ {1− n,−n− 1,−n− 3, . . .}.
1Here and on, different notations 〈·, ·〉 and (·, ·) are used for distributions on Rn
and Sn−1, respectively.
INTERSECTION BODIES 19
Since De(Sn−1) is dense in D′e(Sn−1) and the maps E1−n−α and Eα−1
are weakly continuous from D′e(Sn−1) to S ′(Rn), then (4.1) extends to
all f ∈ D′e(Sn−1).
Regarding the cases excluded in (4.2), we note that if α = 1+ 2ℓ for
some ℓ = 0, 1, . . ., then (4.1) is meaningful if and only if f is orthogonal
to all spherical harmonics of degree 2ℓ. If α = 1 − n − 2ℓ for some
ℓ = 0, 1, . . ., then, according to the spherical harmonic decomposition
j,k fj,kYj,k, j even, formula (4.1) is substituted for the following:
[E2ℓf ]
∧(ξ) = (2π)n
fj,k(−∆)ℓ−j/2Yj,k(i∂) δ(ξ)(4.3)
+2n+2ℓπn/2E−n−2ℓM
1−n−2ℓ
fj,kYj,k
where −∆ is the Laplace operator, ∂ = (∂/∂ξ1, . . . , ∂/∂ξn), and δ(ξ)
is the delta function. It is worth noting that for α = 1, 3, 5, . . ., the
distribution [E1−n−αf ]
∧ can also be understood in the regularized sense
without any orthogonality assumptions. However, such regularization
does not preserve homogeneity; see [Sa1], [Sa3].
Our main concern is positivity and positive definiteness of even ho-
mogeneous distributions. The reader is referred to [GV] for the general
theory. A distribution F ∈ S ′(Rn) is positive if 〈F, φ〉 ≥ 0 for all non-
negative φ ∈ S(Rn). A similar definition holds for distributions on the
sphere and on Rn \ {0}. A distribution F ∈ S ′(Rn) is positive definite
if F̂ is positive. For our purposes, it is important to know, which even
homogeneous distributions are positive definite. Let us rewrite (4.1)
and (4.2) with 1− n− α replaced by −λ. We have
(4.4) [E−λf ]
∧ = 2n−λπn/2Eλ−nM
1+λ−nf,
(4.5) λ /∈ Λ0, Λ0 = {n, n + 2, n+ 4 . . .} ∪ {0,−2,−4, . . .}.
Theorem 4.1. Let λ ∈ R \ Λ0, f ∈ D′e(Sn−1).
(i) If λ < 0 and E−λf is a positive definite distribution, then f = 0.
(ii) For all λ ∈ R \ Λ0, the following statements are equivalent:
(a) [E−λf ]
∧ is a positive distribution on Rn \{0} (for λ > 0, this can
be replaced by “E−λf is a positive definite distribution on R
(b) M1+λ−nf ∈ Me+(Sn−1);
(c) f =M1−λµ for some measure µ ∈ Me+(Sn−1).
Furthermore, for any real λ 6= 0,−2,−4, . . ., and any i = 1, 2, . . . , n−1,
(c) is equivalent to
(d) Rif = R
n−i,⊥µ for some measure µ ∈ Me+(Sn−1).
20 BORIS RUBIN
Proof. (i) Choose φ(x) = exp(−|x|m) pt,θ(x/|x|), where m ∈ 2N and
pt,θ(·) is the Poisson kernel
(4.6) pt,θ(u) =
1− t2
(1− 2tu · θ + t2)n/2
, 0 < t < 1; u, θ ∈ Sn−1.
Then 〈Eλ−nM1+λ−nf, φ〉 = cλ(ΠtM1+λ−nf)(θ), where
cλ = a.c.
rλ−1 exp(−rm) dr = m−1Γ(λ/m)
and (ΠtM
1+λ−nf)(θ) is the Poisson integral of M1+λ−nf . If E−λf is a
positive definite distribution, then, by (4.4), Eλ−nM
1+λ−nf is a positive
distribution. On the other hand, if λ < 0 and m > −λ, then cλ < 0.
Hence 〈Eλ−nM1+λ−nf, φ〉 can be non-negative for every non-negative
φ ∈ S(Rn) only if (ΠtM1+λ−nf)(θ) = 0 for every 0 < t < 1 and
θ ∈ Sn−1. The latter implies M1+λ−nf = 0, which is equivalent to
f = 0 because M1+λ−n is injective; see Lemma 3.2.
(ii) Let [E−λf ]
∧ be a positive distribution on Rn \{0}. It means that
for every φ ∈ S(Rn) such that φ ≥ 0 and 0 /∈ suppφ, 〈[E−λf ]∧, φ〉 ≥ 0
or, by (4.4), 〈Eλ−nM1+λ−nf, φ〉 ≥ 0. Choose φ(x) = ψ(|x|)ω(x/|x|),
where ω ∈ D(Sn−1), ω ≥ 0, and ψ is a smooth non-negative function
such that
rα+n−2ψ(r)dr = 1 and 0 /∈ suppψ. Then
〈Eλ−nM1+λ−nf, φ〉 = (M1+λ−nf, ω) ≥ 0,
and therefore, M1+λ−nf ∈ Me+(Sn−1); see Theorem 9.1.
Conversely, let µ = M1+λ−nf ∈ Me+(Sn−1) and let φ ∈ S(Rn);
φ ≥ 0. In the case λ < 0 we additionally assume 0 /∈ suppφ. By (4.4),
〈[E−λf ]∧, φ〉 = 2n−λπn/2 〈Eλ−nµ, φ〉
= 2n−λπn/2
rλ−1dr
φ(rθ)dµ(θ) ≥ 0.
This proves equivalence of (a) and (b). Equivalence of (b) and (c)
follows from Lemma 3.2.
Let us prove the equivalence of (c) and (d). If Rif = R
n−i,⊥µ,
µ ∈ Me+(Sn−1), then, by (3.15),
(f, R∗iϕ) = (Rif, ϕ) = (R
n−i,⊥µ, ϕ) = (µ,
Ri−λn−iϕ
= c−1(µ,M1−λR∗iϕ), ϕ ∈ D(Gn,i).
Since any function ω ∈ De(Sn−1) can be expressed as ω = R∗iϕ for
some ϕ ∈ D(Gn,i) (see Lemma 3.7), this gives (f, ω) = c−1(µ,M1−λ, ω)
which is (c). Conversely, let f = M1−λµ, µ ∈ Me+(Sn−1), that is,
(f, ω) = (µ,M1−λ, ω) for every ω ∈ De(Sn−1). Choose ω = R∗iϕ, ϕ ∈
INTERSECTION BODIES 21
D(Gn,i). Then, as above, (f, R∗iϕ) = (µ,M1−λR∗iϕ) = c (µ,
Ri−λn−iϕ
which gives (d). �
5. λ-intersection bodies
5.1. Definitions and comments. We remind that Kn is the set of
all origin-symmetric star bodies K in Rn, n ≥ 2; ρK and || · ||K are the
radial function and the Minkowski functional of K. The following defi-
nitions and statements are motivated by Theorem 4.1 and the previous
consideration. Let λ be a real number,
(5.1) sλ =
1 if λ > 0, λ 6= n, n+ 2, n+ 4, . . . ,
Γ(λ/2) if λ < 0, λ 6= −2,−4, . . . .
The values λ = 0, n, n + 2, n + 4, . . . will not be considered in the
following, but values λ = −2,−4, . . . will be included. They become
meaningful if we change normalization. For λ 6= 0, n, n + 2, n + 4 . . . ,
let Inλ be the set of bodies K ∈ Kn, for which there is a measure
µ ∈ Me+(Sn−1) such that sλρK = M1−λµ if λ 6= −2ℓ, ℓ ∈ N, and
ρK = M̃
1−λµ ≡ M̃1+2ℓµ, otherwise. The equality sλρK = M1−λµ
means that for any ϕ ∈ D(Sn−1),
ρkK(θ)ϕ(θ) dθ =
(M1−λϕ)(θ) dµ(θ),
where for λ ≥ 1, (M1−λϕ)(θ) is understood in the sense of analytic
continuation. We remind the notation
Λ0 = {n, n+ 2, n+ 4 . . .} ∪ {0,−2,−4, . . .}.
Theorem 5.1. For λ ∈ R\Λ0, the following statements are equivalent:
(a) K ∈ Inλ ;
(b) The Fourier transform [sλ || · ||−λK ]∧ is a positive distribution on
n\{0} (for λ > 0, this can be replaced by “|| · ||−λK is a positive definite
distribution on Rn”);
(c) sλM
1+λ−nρλK ∈ Me+(Sn−1);
The theorem is an immediate consequence of Theorem 4.1 if the lat-
ter is applied to f = sλρ
K . Another useful characterization is provided
by Theorem 4.1 (d).
Theorem 5.2. Let λ ∈ R \ Λ0. If K ∈ Inλ , then for every i ∈
{1, 2, . . . , n−1} there is a measure µ ∈ Me+(Sn−1) such that sλRiρλK =
Ri−λn−i,⊥µ. Conversely, if
sλRiρ
K = R
n−i,⊥µ
for some i ∈ {1, 2, . . . , n−1} and some µ ∈ Me+(Sn−1), then K ∈ Inλ .
22 BORIS RUBIN
Although Inλ was called “the set of bodies”, the definition of this set
is purely analytic and extra work is needed to understand what bodies
(if any) actually constitute the class Inλ .
The following comments will be helpful.
1. The case λ > n is not so interesting, because by Theorem 5.1(c),
Inλ is either empty (if Γ((n − λ)/2) < 0) or coincides with the whole
class Kn (if Γ((n− λ)/2) > 0).
2. The case λ ∈ (0, n) agrees with the concept of isometric embed-
ding of the space (Rn, || · ||K) into L−p, p = λ; see Introduction. In the
framework of this concept, all bodies K ∈ Inλ can be regarded as “unit
balls of n-dimensional subspaces of L−λ”.
3. If K ∈ Inλ , where λ < 0 (one can replace λ by = −p, p > 0), then
||u||pK =
|θ · u|p dµ(θ)
for some µ ∈ Me+(Sn−1). This is the well known Lévy representation,
characterizing isometric embedding of the space (Rn, || · ||K) into Lp;
see Lemma 6.4 in [K4]. Statement (b) in Theorem 5.1 agrees with
Theorem 1.9. Keeping this terminology, we can state the following
Proposition 5.3. Let p > −n, p 6= 0. Then (Rn, || · ||K) embeds
isometrically in Lp if and only if K ∈ In−p.
4. If λ = k ∈ {1, 2, . . . , n−1}, then Inλ = Ink coincides with the class
of k-intersection bodies; see Definition 1.7 and Theorem 1.8. Theorems
5.1 and 5.2 provide new characterizations of this class.
These comments inspire the following
Definition 5.4. Let λ < n, λ 6= 0. A body K ∈ Kn is said to be a
λ-intersection body if K ∈ Inλ , or, in other words, if there is a measure
µ ∈ Me+(Sn−1) such that sλρλK = M1−λµ if λ 6= −2ℓ, ℓ ∈ N, and
ρ−2ℓK = M̃
1+2ℓµ, otherwise.
The result of Theorem 5.2 for λ = i = k can serve as an alternative
definition of k-intersection bodies in terms of Radon transforms. This
definition agrees with Definition 1.6 and mimics Definition 1.2.
Definition 5.5. Let k ∈ {1, 2, . . . , n − 1}. A body K ∈ Kn is a k-
intersection body if there is a non-negative measure µ on Sn−1 such
(5.2) (Rkρ
K)(ξ) = (Rn−kµ)(ξ
⊥), ξ ∈ Gn,k.
INTERSECTION BODIES 23
Equality (5.2) is understood in the weak sense according (2.5). Namely,
for ϕ ∈ C(Gn,k) and ϕ⊥(η) = ϕ(η⊥), η ∈ Gn,n−k, (5.2) means
(5.3)
K)(ξ)ϕ(ξ) dξ =
(R∗n−kϕ
⊥)(θ) dµ(θ).
5.2. λ-intersection bodies of star bodies and closure in the ra-
dial metric. As we mentioned in Introduction, the class of intersection
bodies, which coincides with Inλ when λ = 1, is the closure in the ra-
dial metric of the class of intersection bodies of star bodies. Below
we extend this result to all λ < n, λ 6= 0, in the framework of the
unique approach. We remind (see Definition 1.6) that K ∈ Kn is a
k-intersection body of a body L ∈ Kn and write K = IBk(L) if
(5.4) volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) ∀ξ ∈ Gn,k.
Let IBk,n be the set of all bodies K ∈ Kn satisfying (5.4) for some
L ∈ Kn.
How can we extend the purely geometric property (5.4) to non-
integer values of k? To this end, we first express (5.4) in terms of
the generalized cosine transforms (2.9).
Lemma 5.6. If K = IBk(L) is infinitely smooth, then
(5.5) ρn−kL =cM
1−n+kρkK , ρ
−1M1−kρn−kL ,
c = πk−n/2(n− k)/k.
Proof. We make use of (3.13), where we set i = k, α = 1 − n + k and
f = ρkK . By (3.1), this gives
(5.6) Rkρ
K = c̃Rn−k,⊥M
1−n+kρkK , c̃ =
πk−n/2 σn−k−1
On the other hand, if K = IBk(L) is infinitely smooth, then, according
to (5.4) and the equality
(5.7) volk(K ∩ ξ) =
K)(ξ),
we have
(5.8) Rkρ
k σn−k−1
(n− k) σk−1
Rn−k,⊥ρ
Comparing (5.6) and (5.8), owing to injectivity of the Radon transform,
we obtain the first equality in (5.5). The second equality follows from
the first one by (3.5). �
24 BORIS RUBIN
Equalities (5.5) are extendable to non-integer values of k. We denote
cλ,n = π
λ−n/2(n−λ)/λ,
and let sλ be defined by (5.1).
Definition 5.7. Let λ < n, λ 6= 0; K,L ∈ Kn. We say that K is a
λ-intersection body of L and write K = IBλ(L) if sλρλK=c−1λ,nM1−λρ
in the case λ 6= −2ℓ, ℓ ∈ N, and ρ−2ℓK = M̃1+2ℓρ
L , otherwise. The
set of all λ-intersection bodies of star bodies will be denoted by IBλ,n.
We also denote
(5.9) IB∞λ,n={K ∈ IBλ,n : ρK ∈ De(Sn−1)}.
By (3.5), equality sλρ
K = c
1−λρn−λL is equivalent to ρ
sλ cλ,nM
1−n+λρλK . Both equalities are generally understood in the sense
of distributions, for instance,
K , ϕ) = c
λ,n(ρ
1−λϕ), ϕ ∈ D(Sn−1).
If K (or L) is smooth, then sλρ
K(θ)=c
λ,n(M
1−λρn−λL )(θ) pointwise for
every θ∈Sn−1.
Theorem 5.8. Let λ < n, λ 6= 0. If λ 6= −2ℓ, ℓ ∈ N, then the class
Inλ of λ-intersection bodies is the closure of the classes IBλ,n and IB∞λ,n
of λ-intersection bodies of star bodies in the radial metric:
(5.10) Inλ = cl IBλ,n = cl IB∞λ,n.
If λ = −2ℓ, ℓ ∈ N, then Inλ ⊂ cl IBλ,n = cl IB∞λ,n.
Proof. STEP 1. We first prove that Inλ ⊂ cl IB∞λ,n. Let K ∈ Inλ , i.e.,
(a) sλρ
1−λµ, µ ∈ Me+(Sn−1), if λ 6= −2ℓ, ℓ ∈ N, and
(b) ρ−2ℓK = M̃
1+2ℓµ, otherwise.
Our aim is to define a sequence Kj ∈ IB∞λ,n such that ρKj → ρK in the
C-norm. Consider the Poisson integral Πtρ
K (see (2.1)), that converges
to ρλK in the C-norm when t→ 1. In the case (a), for any test function
ω ∈ D(Sn−1) we have
K , ω) = (ρ
K ,Πtω) = s
λ (µ,M
1−λΠtω) = s
1−λΠtµ, ω).
Similarly, in the case (b), we have a pointwise equality (Πtρ
K )(θ) =
(M̃1+2ℓΠtµ)(θ), θ ∈ Sn−1. Choose Kj so that ρλKj = Πtjρ
K , where tj is
a sequence in (0, 1) approaching 1. Clearly, Kj converges to K in the
radial metric. Moreover, Kj ∈ IB∞λ,n, because ρλKj = s
1−λρn−λLj
and ρ−2ℓKj = M̃
1+2ℓρn+2ℓLj , where the bodies Lj are defined by ρ
cλ,nΠtjµ in the case (a), and ρ
= Πtjµ in the case (b), respectively.
INTERSECTION BODIES 25
Conversely, let K ∈ cl IB∞λ,n, λ 6= −2,−4, . . . . It means that there
is a sequence of Kj ∈ IB∞λ,n such that lim
||ρK − ρKj ||C(Sn−1) = 0 and
= c−1λ,nM
1−λρn−λLj , ρLj ∈ De+(S
n−1). If j → ∞, then for every
ω ∈ D(Sn−1),
(5.11) sλ(ρ
,M1−n+λω) → sλ(ρλK ,M1−n+λω)=sλ(M1−n+λρλK , ω).
The right-hand side of (5.11) is non-negative, because by (3.5), for
every j and every ω ∈ De+(Sn−1),
,M1−n+λω) = c−1λ,n(M
1−λρn−λLj ,M
1−n+λω) = c−1λ,n(ρ
, ω) ≥ 0.
By Theorem 9.1, it follows that sλM
1−n+λρλK is a non-negative mea-
sure. We denote it by µ. By (3.5), for any ω ∈ D(Sn−1),
K , ω) = sλ(M
1−n+λρλK ,M
1−λω) = (µ,M1−λω) = (M1−λµ, ω),
i.e., K∈Inλ . This gives IB∞λ,n⊂Inλ and, by above, Inλ =cl IB∞λ,n.
STEP 2. It remains to prove that cl IB∞λ,n = cl IBλ,n. Since IB∞λ,n ⊂
IBλ,n, then cl IB∞λ,n ⊂ cl IBλ,n. To prove the opposite inclusion, let
K ∈ cl IBλ,n and consider the case λ 6= −2,−4, . . . . We have to show
that there is a sequence of smooth bodies Kj, which converges to K in
the radial metric and such that sλρ
= c−1λ,nM
1−λρn−λLj for some bodies
Lj ∈ Kn. Since K ∈ cl IBλ,n, there is a sequence K̃j ∈ Kn such that
||ρK̃j −ρK ||C(Sn−1) = 0 and sλρ
= c−1λ,nM
1−λρn−λ
for some bodies
L̃j ∈ Kn. We define smooth bodies Kj and Lj by
ρλKj = Π1−1/jρ
, ρn−λLj = Π1−1/jρ
where Π1−1/j stands for the Poisson integral with parameter 1 − 1/j.
Since operators Π1−1/j andM
1−λ commute, then sλρ
=c−1λ,nM
1−λρn−λLj ,
and therefore, Kj ∈ IB∞λ,n. On the other hand, by the properties of the
Poisson integral [SW],
|ρλKj − ρ
K | ≤ |Π1−1/jρλK̃j − Π1−1/jρ
K |+ |Π1−1/jρλK − ρλK | → 0
as j → ∞. It means, that K ∈ cl IB∞λ,n or cl IBλ,n ⊂ cl IB∞λ,n. Hence,
by above, cl IBλ,n = cl IB∞λ,n. For λ = −2,−4, . . . , the argument is
similar. �
Remark 5.9. If λ = −2,−4, . . . , we cannot prove the coincidence of
Inλ and cl IB∞λ,n, because the proof of the embedding cl IB∞λ,n ⊂ Inλ
relies heavily on the fact that M1−λ is an isomorphism of De(Sn−1). If
λ = −2,−4, . . . , this is not so, and the operator M̃1−λ has a nontrivial
kernel, which consists of spherical harmonics of degree > 2ℓ; see [R1]
for details.
26 BORIS RUBIN
It is interesting to translate Theorem 5.8 for λ = −p, p > 0, into
the language of isometric embeddings. Ignoring a non-important pos-
itive constant factor and using polar coordinates, one can replace the
equalities sλρ
K = c
1−λρn−λL and ρ
K = M̃
1+2ℓρn+2ℓL in Definition
5.7 by
(5.12) ||u||pK =
|x · u|p dx, u ∈ Sn−1.
Corollary 5.10.
(i) A unit ball of every n-dimensional subspace of Lp, can be approxi-
mated in the radial metric by bodies K, defined by (5.12), where L ∈ Kn
has a C∞ boundary.
(ii) If, moreover, p 6= 2, 4, . . . , then the set of unit balls of all n-
dimensional subspaces of Lp, can be identified with the closure in the
radial metric of the set of bodies K satisfying (5.12) for some smooth
body L ∈ Kn (one can also consider arbitrary bodies L ∈ Kn).
5.3. Central sections of λ-intersection bodies. It is known, that
a cross-section K ∩ η of a body K ∈ Ink by any m-dimensional central
plane η is a k-intersection body in η provided 1 ≤ k < m < n. This
fact was established in [Mi1, Proposition 3.17] by using Theorem 1.8
and a certain approximation procedure. Below we present more general
results, including sections of k-intersection bodies of star bodies and
the case of non-integer k = λ. These results are consequences of the
restriction theorems from Section 3.3.
Theorem 5.11. Let 1 ≤ k < m < n, η ∈ Gn,m. If K = IBk(L) in Rn,
then K ∩ η = IBk(L̃) in η, where the body L̃ is defined by
(5.13) ρm−k
(u) = ck,m,n
Sn−1∩(η⊥⊕Ru)
ρn−kL (w)|u · w|
m−k−1 dw,
u ∈ Sn−1 ∩ η, ck,m,n =
(m− k) σn−m
2(n− k)
Proof. By (5.7) and (3.21) (with f = ρn−kL ),
volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) =
σn−k−1
(Rn−kρ
L )(ξ
c σn−k−1
(Rm−kT
L )(ξ
⊥ ∩ η)(5.14)
σm−k−1
(Rm−kρ
)(ξ⊥ ∩ η) = volm−k(L̃ ∩ ξ⊥),
as desired. �
INTERSECTION BODIES 27
Theorem 5.11 has the following generalization.
Theorem 5.12. Let 1 < m < n, η ∈ Gn,m and suppose that λ < m,
λ 6= 0. If K = IBλ(L) in Rn, then K ∩ η = IBλ(L̃) in η, where the
body L̃ is defined by
(5.15) ρm−λ
(u) = c̃
Sn−1∩(η⊥⊕Ru)
ρn−λL (w)|u · w|
m−λ−1 dw,
u ∈ Sn−1 ∩ η, c̃ =
(m− λ) σn−m
2(n− λ)
if λ 6= −2ℓ, ℓ ∈ N,
π(m−n)/2 σn−m/2 otherwise.
Moreover, if K ∈ Inλ in Rn, then K ∩ η ∈ Imλ in η.
Proof. Let λ 6= −2ℓ, ℓ ∈ N, and let θ ∈ Sn−1 ∩ η. By Definition 5.7,
K = c
1−λρn−λL , and Theorem 3.12 (with f = sλρ
K and g =
c−1λ,nρ
L ) yields
K(θ) = (M
Sn−1∩η
T λη [c
L ])(θ) = c
λ,m(M
Sn−1∩η
)(θ),
where ρm−λ
= c T λη ρ
L , c = π
(n−m)/2(m − λ)/(n − λ). By Definition
5.7 and (3.20), we are done. If λ = −2ℓ, ℓ ∈ N, then, as above,
ρ−2ℓK (θ) = (M̃
Sn−1∩η
T−2ℓη ρ
L )(θ) = (M
Sn−1∩η
ρm+2ℓ
where ρm+2ℓ
= T−2ℓη ρ
L . This gives (5.15).
Furthermore, if K ∈ Inλ , λ 6= −2ℓ, ℓ ∈ N, then, by Definition 5.4,
K = M
1−λµ, µ ∈ Me+(Sn−1). Hence, by Theorem 3.12, there is
a measure ν ∈ Me+(Sn−1 ∩ η) such that the restriction of sλρλK onto
Sn−1∩η is represented as sλρλK =M1−λSn−1∩ην. It means that K∩η ∈ I
in η. In the case λ = −2ℓ, ℓ ∈ N, the argument is similar. �
6. Examples of λ-intersection bodies
The definition of the classes Inλ and IBλ,n and all known characteri-
zations are purely analytic. Unlike the case λ = 1, when an intersection
body of a star body is explicitly defined by a simple geometric proce-
dure, it is not clear how can we construct λ−intersection bodies in the
general case. Below we give some examples, when the radial function
of a λ−intersection body can be explicitly determined. These examples
utilize the generalized cosine transforms.
Example 6.1. Let λ < 1, λ 6= 0. This case is the simplest. Indeed,
given a non-negative measure µ on Sn−1, the relevant λ−intersection
28 BORIS RUBIN
body can be directly constructed by the formula ρλK = M
1−λµ, if λ 6=
−2ℓ, ℓ ∈ N, and ρ−2ℓK = M̃1+2ℓµ, otherwise. In other words (cf. (2.11)),
(6.1) ρλK(u) =
|θ · u|−λ dµ(θ).
This fact (with λ replaced by −p) is a reformulation of Theorem 6.17
from [K4], which was stated in the language of isometric embeddings
and relies on the P. Lévy characterization; see also Lemma 6.4 and
Theorem 4.11 in [K4].
Example 6.2. If n − 3 ≤ λ < n, λ > 0, then Inλ includes all origin-
symmetric convex bodies in Rn.
This fact is due to Koldobsky [K4, Corollary 4.9]. It can be proved
using a slight modification of the argument from [R3, Sec. 7] as follows.
By Theorem 5.1 (c), it suffices to check that for any o.s. convex body
K,M1+λ−nρλK ∈ Me+(Sn−1). For λ ≥ n−1, this is obvious. To handle
the case n− 3 ≤ λ < n− 1, suppose first that K is infinitely smooth.
Using polar coordinates, for Reα > 0, we can write
(6.2) (Mαρα+n−1K )(u) = (α + n− 1) γn(α)
|x · u|α−1 dx.
Then M1+λ−nρλK can be realized as analytic continuation (a.c.) at α =
1 + λ− n of the right-hand side of (6.2). The latter can be written as
I(α) = 2(α+ n− 1)γn(α)
tα−1AK,u(t) dt,
AK,u(t) = voln−1(K∩{tu+u⊥}). Taking analytic continuation (see [GS,
Chapter 1]), for −2 < α < 0 (which is equivalent to n−3 ≤ λ < n−1)
we get
a.c.I(α) = c1
tα−1[AK,u(t)−AK,u(0)] dt.
Similarly, a.c.I(α) at α = −2 (which corresponds to λ = n − 3) is
K,u(0). Following [GS], one can easily check that constants c1 and
c2 are negative. Since K is convex, both analytic continuations are
positive, and thereforeM1+λ−nρλK > 0. If K is an arbitrary o.s. convex
body, we approximate it in the radial metric by smooth o.s. convex
bodies Kj. Then for any test function ω ∈ D+(Sn−1), by the previous
step we have
(M1+λ−nρλK , ω) = (ρ
1+λ−nω) = lim
(ρλKj ,M
1+λ−nω)
= lim
(M1+λ−nρλKj , ω) ≥ 0.
INTERSECTION BODIES 29
Hence, by Theorem 9.1, M1+λ−nρλK is a non-negative measure and the
proof is complete.
Example 6.3. If ρλK =
Ri−λn−iν for some ν ∈ M+(Gn,n−i) and λ ≤ i < n,
then K ∈ Inλ .
Indeed, for any test function ω ∈ D(Sn−1), by (3.12) (with α = 1−λ)
we have
(ρλK , ω) = (
Ri−λn−iν, ω) = (ν, R
n−iω) = (ν
⊥, Ri−λn−i,⊥ω)
= c−1(ν⊥, RiM
1−λω) = c−1(R∗i ν
⊥,M1−λω), c =
2π(i−1)/2
It means that for 0 < λ ≤ i < n and ν ∈ M+(Gn,n−i),
(6.3) ρλK =
Ri−λn−iν ⇐⇒ {ρλK =M1−λµ, µ = c−1R∗i ν⊥}.
By Definition 5.4, this gives the result. The particular case λ = i
implies the embedding into Ini of the Zhang’s class Znn−i; see Definition
1.5. This embedding was proved in [K3] and [Mi1] in a different way;
see also [Mi2], where it is proved that Znn−i is a proper subset of Ini if
2 ≤ i ≤ n− 2.
Example 6.4. If 0 < (i− 1)/2 < λ ≤ i < n and ρλK = M i−λµ for some
µ ∈ M+(Sn−1), then K ∈ Inλ .
Indeed, by Lemma 3.4 (with α = i− λ, β = 1− λ), ρλK =M i−λµ =
M1−λAi−λ,1−λ, where Ai−λ,1−λ is an integral operator which preserves
positivity provided i− λ > 1− λ > 1− n, (i− λ) + (1− λ) < 2. This
is just our case.
Example 6.5. One can construct bodies K ∈ Inλ from bodies L ∈ Inδ
by the formula ρK = ρ
L provided 0 < δ < λ < n.
Indeed, by Definition 5.4, there is a measure µ ∈ M+(Sn−1) so that
ρδL = M
1−δµ. Then, by Lemma 3.4 (with α = 1 − δ, β = 1 − λ),
ρλK = ρ
L = M
1−δµ = M1−λA1−δ,1−λµ, and we are done. This example
generalizes the corresponding result from [Mi1, p. 533, Statement (c)],
which was obtained in a different way for the case, when λ and δ are
integers.
Example 6.6. Let
(6.4) Bnq = {x ∈ Rn : ||x||q =
|xk|q
≤ 1}.
If 0 < q ≤ 2, then Bnq ∈ Inλ for all λ ∈ (0, n). If 2 < q <∞, λ ∈ (0, n),
then Bnq ∈ Inλ if and only if λ ≥ n− 3.
30 BORIS RUBIN
Both statements are due to Koldobsky. The first one follows from
the fact that for 0 < q ≤ 2 the Fourier transform of ||x||−λq is a positive
S ′-distribution (see Lemmas 3.6 and 2.27 in [K4]). The second state-
ment is a reformulation of Theorem 4.13 from [K4]. The “if” part is a
consequence of Example 6.2.
7. (q, ℓ)-balls
In this section we consider one more example, which resembles Ex-
ample 6.6, but does not fall into its scope and requires a separate
consideration. Let
x = (x′, x′′) ∈ Rn, x′ ∈ Rn−ℓ =
Rej , x
′′ ∈ Rℓ =
j=n−ℓ+1
Rej ,
e1, . . . , εn being coordinate unit vectors. Consider the (q, ℓ)-ball
(7.1) Bnq,ℓ = {x : ||x||q,ℓ = (|x′|q + |x′′|q)1/q ≤ 1}, q > 0.
We wonder, for which triples (q, ℓ, n), Bnq,ℓ is a λ-intersection body. To
study this problem, we need some preparation. Consider the Fourier
integral
(7.2) γq,ℓ(η) =
e−|y|
eiy·η dy, η ∈ Rℓ, q > 0.
The function γq,ℓ(η) is uniformly continuous on R
ℓ and vanishes at
infinity.
Lemma 7.1. If 0 < q ≤ 2, then γq,ℓ(η) > 0 for all η ∈ Rℓ.
Proof. (Cf. [K4, p. 44, for ℓ = 1]). For η = 0, the statement is obvious.
It is known (see, e.g., [SW]), that
(7.3) [e−t|· |
]∧(η) = πℓ/2t−ℓ/2e−|η|
2/4t, t > 0.
This gives the result for q = 2. Let 0 < q < 2. By Bernstein’s theorem
[F, Chapter 18, Sec. 4], there is a non-negative finite measure µq on
[0,∞) so that e−zq/2 =
e−tz dµq(t), z ∈ [0,∞). Replace z by |y|2 to
(7.4) e−|y|
e−t|y|
dµq(t).
Then (7.3) yields
γq,ℓ(η) =
eiy·ηdy
e−t|y|
dµq(t) =
dµq(t)
eiy·ηe−t|y|
= πℓ/2
t−ℓ/2e−|η|
2/4t dµq(t) > 0.
INTERSECTION BODIES 31
The Fubini theorem is applicable here, because, by (7.4),
|eiy·η|dy
e−t|y|
dµq =
e−|y|
dy <∞.
Our next concern is the behavior of γq,ℓ(η) when |η| → ∞. If q
is even, then e−|·|
is a Schwartz function and therefore, γq,ℓ is infin-
itely smooth and rapidly decreasing. In the general case, we have the
following.
Lemma 7.2. For any q > 0,
(7.5) lim
|η|→∞
|η|ℓ+qγq,ℓ(η) = 2ℓ+qπℓ/2−1Γ(1+ q/2)Γ((ℓ+ q)/2) sin(πq/2).
Proof. For ℓ = 1, this statement can be found in [PS, Chapter 3, Prob-
lem 154] and in [K4, p. 45]. In the general case, the proof is more
sophisticated and relies on the properties of Bessel functions. By the
well-known formula for the Fourier transform of a radial function (see,
e.g., [SW]), we write γq,ℓ(η) = I(|η|), where
I(s) = (2π)ℓ/2s1−ℓ/2
rℓ/2Jℓ/2−1(rs) dr
= (2π)ℓ/2s−ℓ
[(rs)ℓ/2Jℓ/2(rs)] dr.
Integration by parts yields
I(s) = q(2π)ℓ/2s−ℓ/2
rℓ/2+q−1Jℓ/2(rs) dr.
Changing variable z = sqrq, we obtain
sℓ+qI(s) = (2π)ℓ/2A(s1/q), A(δ) =
e−zδzℓ/2qJℓ/2(z
1/q) dz.
We actually have to compute the limit A0 = lim
A(δ). To this end, we
invoke Hankel functions H
ν (z), so that Jν(z) = ReH
ν (z) if z is real
[Er]. Let hν(z) = z
ν (z). This is a single-valued analytic function
in the z-plane with cut (−∞, 0]. Using the properties of the Bessel
functions [Er], we get
(7.6) lim
hν(z) = 2
νΓ(ν)/πi,
(7.7) hν(z) ∼
2/π zν−1/2eiz−
(ν+ 1
), z → ∞.
Then we write A(δ) as A(δ) = Re
e−zδhℓ/2(z
1/q) dz and change the
line of integration from [0,∞) to ℓθ = {z : z = reiθ, r > 0} for
32 BORIS RUBIN
small θ < πq/2. By Cauchy’s theorem, owing to (7.6) and (7.7), we
obtain A(δ) = Re
e−zδhℓ/2(z
1/q) dz. Since for z = reiθ, hℓ/2(z
1/q) =
O(1) when r = |z| → 0 and hℓ/2(z1/q) = O(r(ℓ−1)/2qe−r
1/q sin(θ/q)) as
r → ∞, by the Lebesgue theorem on dominated convergence, we get
A0 = Re
hℓ/2(z
1/q) dz. To evaluate the last integral, we again use
analyticity and replace ℓθ by ℓπq/2 = {z : z = reiπq/2, r > 0} to get
A0 = Re
eiπq/2
hℓ/2(r
1/qeiπ/2) dr
To finalize calculations, we invoke McDonald’s function Kν(z) so that
hν(z) = z
νH(1)ν (z) = −
(ze−iπ/2)νKν(ze
−iπ/2).
This gives
sin(πq/2)
sℓ/2+q−1Kℓ/2(s) ds.
The last integral can be explicitly evaluated by the formula 2.16.2 (2)
from [PBM], and we obtain the result. �
Now we can proceed to studying (q, ℓ)-balls Bnq,ℓ; see (7.1). There is
an intimate connection between geometric properties of the balls Bnq,ℓ
and the Fourier transform of the power function || · ||pq,ℓ. The case q = 2
is well-known and associated with Riesz potentials; see, e.g., [St]. The
relevant case of ℓnq -balls, which agrees with ℓ = 1 was considered in
Example 6.6.
Lemma 7.3. Let q > 0, ξ = (ξ′, ξ′′) ∈ Rn, γq,ℓ(ξ′′) and γq,n−ℓ(ξ′) be
the functions of the form (7.2). We define
(7.8) hp,q,ℓ(ξ) =
Γ(−p/q)
tn+p−1 γq,n−ℓ(ξ
′t) γq,ℓ(ξ
′′t) dt.
(i) Let ξ′ 6= 0 and ξ′′ 6= 0. If q is even, then the integral (7.8) is abso-
lutely convergent for all p > −n. Otherwise, it is absolutely convergent
when −n < p < 2q. In these cases, hp,q,ℓ (ξ) is a locally integrable
function away from the coordinate subspaces Rℓ and Rn−ℓ.
(ii) If −n < p < 0, then hp,q,ℓ (ξ) ∈ L1loc(Rn)∩S ′(Rn) and (||·||
∧(ξ) =
hp,q,ℓ(ξ) in the sense of S ′-distributions. Specifically, for ϕ ∈ S(Rn),
(7.9) 〈hp,q,ℓ , ϕ̂〉 = (2π)n〈|| · ||pq,ℓ , ϕ〉.
INTERSECTION BODIES 33
Proof. (i) For any 0 < ε < a <∞,
ε<|ξ′|<a
ε<|ξ′′|<a
|hp,q,ℓ (ξ; , ξ′′)| dξ′′
|Γ(−p/q)|
tn+p−1 dt
ε<|ξ′|<a
|γq,n−ℓ (ξ′t)| dξ′
ε<|ξ′′|<a
|γq,ℓ (ξ′′t)| dξ′′
|Γ(−p/q)|
tp−1 dt
tε<|z′|<ta
|γq,n−ℓ (z′)| dz′
tε<|z′′|<ta
|γq,ℓ (z′′)| dz′′
|Γ(−p/q)|
(...) =
|Γ(−p/q)|
(I1 + I2).
The first integral is dominated by
tn+p−1 dt, c = σn−ℓ−1σℓ−1max
|γq,n−ℓ (z′)| max
|γq,ℓ (z′′)|
and is finite for p > −n. The second integral can be estimated by
making use of Lemma 7.2. Specifically, if q is not an even integer, then
I2 ≤ cε
tp−1 dt
|z′|>tε
|z′|n−ℓ+q
|z′′|>tε
|z′′|ℓ+q
tp−2q−1 dt.
If q is even, then γq,ℓ and γq,n−ℓ are rapidly decreasing and I2 ≤
tp−2m−1 dt for any m > 0. This gives what we need.
(ii) If −n < p < 0, the same argument is applicable with ε = 0. In
this case, I2 does not exceed ||γq,n−ℓ||1||γq,ℓ||1
tp−1 dt. The latter is
finite when p < 0, because, by Lemma 7.2, γq,n−ℓ and γq,ℓ are integrable
functions on respective spaces. When ξ → ∞, one can readily check
that hp,q,ℓ (ξ) = O(|ξ|m) for some m > 0, and therefore, hp,q,ℓ ∈ S ′(Rn).
To compute the Fourier transform (|| · ||pq,ℓ)∧(ξ), we replace ||x||
q,ℓ by
the formula
||x||pq,ℓ =
Γ(−p/q)
tp−1 e−|x
′/t|q−|x′′/t|q dt, p < 0,
34 BORIS RUBIN
and note that the Fourier transform of the function x→ e−|x′/t|q−|x′′/t|q
is just γq,n−ℓ (ξ
′t) γq,ℓ (ξ
′′t). Then
〈|| · ||pq,ℓ)
∧ , ϕ̂〉 = (2π)n〈|| · ||pq,ℓ , ϕ〉
(2π)nq
Γ(−p/q)
tp−1 dt
′/t|q−|x′′/t|qϕ(x) dx
Γ(−p/q)
tn+p−1 dt
γq,n−ℓ (ξ
′t) γq,ℓ (ξ
′′t) ϕ̂(ξ) dξ.
Interchange of the order of integration in this argument can be easily
justified using absolute convergence of integrals under consideration.
Theorem 7.4. If 0 < q ≤ 2, 0 < ℓ < n, then Bnq,ℓ is a λ-intersection
body for any 0 < λ < n.
Proof. Owing to Lemma 7.1, the function (7.8) (with p replaced by −λ)
is positive, and therefore, by Lemma 7.3, || · ||−λq,ℓ represents a positive
definite distribution. Now the result follows by Theorem 5.1. �
Consider the case q > 2. In this case Bnq,ℓ is convex, and, owing to
Example 6.2, Bnq,ℓ ∈ Inλ for all n− 3 ≤ λ < n. What about λ < n− 3?
This case is especially intriguing.
Proposition 7.5. If q > 2 and 0 < λ < max(n− ℓ, ℓ)− 2, then || · ||−λq,ℓ
is not a positive definite distribution and therefore, Bnq,ℓ 6∈ Inλ .
Proof. Let 0 < λ < n− ℓ− 2 and suppose the contrary, that Bnq,ℓ ∈ Inλ .
Consider the section of Bnq,ℓ by the (n − ℓ + 1)-dimensional plane η =
Ren ⊕ Rn−ℓ. By Theorem 5.12, Bnq,ℓ ∩ η ∈ In−ℓ+1λ in η, and therefore
||xnen + x′′||λq,ℓ = (|xn|q + |x′′|q)−λ/q
is a positive definite distribution in η. By the second derivative text (see
[K4, Theorem 4.19]) this is impossible if 0 < λ < n− ℓ− 2. A similar
contradiction can be obtained if we assume 0 < λ < ℓ− 2 and consider
the section of Bnq,ℓ by the (ℓ+ 1)-dimensional plane Re1 ⊕ Rℓ. �
Proposition 7.5 can be proved without using the second derivative
text and Theorem 5.12 on sections of λ-intersection bodies; see [R4].
The bounds for λ appear to be the same.
Open problem. Let q > 2, ℓ > 1. Is Bnq,ℓ a λ-intersection body if
max(n− ℓ, ℓ)− 2 < λ < n− 3?
This problem does not occur in the case ℓ = 1 as in Example 6.6.
INTERSECTION BODIES 35
8. The generalized cosine transforms and comparison of
volumes
For 1 < i < n, let voli(·) denote the i-dimensional volume function.
Suppose that i is fixed, and let A and B be o.s. convex bodies in Rn
satisfying
(8.1) voli(A ∩ ξ) ≤ voli(B ∩ ξ) ∀ξ ∈ Gn,i.
Does it follow that
(8.2) voln(A) ≤ voln(B) ?
This question is known as the Generalized Busemann-Petty Problem
(GBP); see [G], [RZ], [Z1].
Theorem 8.1. If GBP (8.1)-(8.2) has an affirmative answer, then
every smooth origin-symmetric convex body with positive curvature in
n is an (n− i)-intersection body.
Proof. Suppose that B is an o.s. convex body in Rn so that the radial
function ρB is infinitely smooth, the boundary of B has a positive curva-
ture and B /∈ Inn−i. By Definition 5.4, there is a function ϕ ∈ De(Sn−1),
which is negative on some open origin-symmetric set Ω ⊂ Sn−1 and
such that ρn−iB = M
1+i−nϕ. We choose a function h ∈ De(Sn−1) so
that h 6≡ 0, h(θ) ≥ 0 if θ ∈ Ω and h(θ) ≡ 0 otherwise. Define an o.s.
smooth body A by ρiA = ρ
B − εM1−ih, ε > 0. If ε is small enough,
then A is convex. Since by (3.12), RiM
1−ih = cR0n−i,⊥h ≥ 0, then
A ≤ RiρiB, which gives (8.1). On the other hand, by (3.5),
(ρn−iB , ρ
B − ρiA) = ε(M1+i−nϕ,M1−ih) = ε(ϕ, h) < 0,
or (ρn−iB , ρ
B) < (ρ
B , ρ
A). By Hölder’s inequality, this implies voln(B) <
voln(A), which contradicts (8.2). �
Remark 8.2. As we noted in Introduction, Theorem 8.1 is not new, and
its proof given in [K3] relies on a sequence of deep facts from functional
analysis. The proof presented above is much more elementary and
constructive. For instance, it allows us to keep invariance properties of
the bodies under control. This advantage was essentially used in our
paper [R4].
Theorem 8.1 and Proposition 7.5 imply the following
Corollary 8.3. Let 1 ≤ ℓ ≤ n/2; i > ℓ+2, B = Bn4,ℓ (see (7.1)). Then
there is a smooth o.s. convex body A in Rn so that (8.1) holds but (8.2)
fails.
36 BORIS RUBIN
Setting ℓ = 1 in this statement, we obtain the well-known Bourgain-
Zhang theorem, which states that GBP has a negative answer when
3 < i < n; see [BZ], [K4], [RZ] on this subject. For i = 2 and i = 3
(n ≥ 5) the GBP is still open. An affirmative answer in these cases
was obtained in [R4] for bodies having a certain additional symmetry.
9. Appendix
Every positive distribution F ∈ S ′(Rn) is given by a tempered non-
negative measure µ, i.e., 〈F, φ〉 =
φ(x)dµ(x); see, e.g., [GV, p.147]).
For convenience of the reader, we present a similar fact for the sphere.
Theorem 9.1. A distribution f ∈ D′(Sn−1) is positive if and only if
there is a measure µ ∈ M+(Sn−1) such that
(f, ϕ) =
ϕ(θ)dµ(θ) ∀ϕ ∈ D(Sn−1).
Proof. This statement is known, however, we could not find precise ref-
erence and decided to give a proof for convenience of the reader. The
“if” part is obvious. To prove the “ only if” part, we write a test func-
tion ϕ ∈ D(Sn−1) as a sum ϕ = ϕ1+iϕ2, where ϕ1 = Reϕ, ϕ2 = Imϕ.
Since −||ϕ||C(Sn−1) ≤ ϕj ≤ ||ϕ||C(Sn−1), j = 1, 2, and f is positive, then
−(f, 1) ||ϕ||C(Sn−1) ≤ (f, ϕj) ≤ (f, 1) ||ϕ||C(Sn−1),
and therefore, |(f, ϕ)| ≤ |(f, ϕ1)|+ |(f, ϕ2)| ≤ 2(f, 1) ||ϕ||C(Sn−1). Since
D(Sn−1) is dense in C(Sn−1), then f extends as a linear continuous
functional f̃ on C(Sn−1) and, by the Riesz theorem, there is a mea-
sure µ on Sn−1 such that (f̃ , ω) =
ω(θ)dµ(θ) for every ω ∈
C(Sn−1). In particular, (f, ϕ) = (f̃ , ϕ) =
ϕ(θ)dµ(θ) for every
ϕ ∈ D(Sn−1). By taking into account that every non-negative function
ω ∈ C(Sn−1) can be uniformly approximated by non-negative functions
ϕk ∈ D(Sn−1) (for instance, by Poisson integrals of ω), we get
ω(θ)dµ(θ) = lim
ϕk(θ)dµ(θ) = lim
(f, ϕk) ≥ 0.
The latter means that µ is non-negative. �
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Department of Mathematics, Louisiana State University, Baton Rouge,
LA, 70803 USA
E-mail address : borisr@math.lsu.edu
