# Busemann-Petty Problems for QuasiLp Intersection Bodies

Authors: Yanping Zhou; Shanhe Wu
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We investigate theLp dual geominimal surface area and volume forms of Busemann-Petty problems for the quasi Lp intersection bodies and establish some new geometric inequalities. Our results provide a significant complement to the researches on Busemann-Petty problems for intersection bodies.


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## Body

## 1. Introduction and Main Results

LetSn-1 denote the unit sphere in Euclidean space Rn. If K is a compact star-shaped (about the origin) set in Rn, then its radial function, ρK=ρ(K,·):Rn∖{0}→[0,∞), is defined by (see [1, 2]) (1)ρK,u=maxλ≥0:λu∈K,u∈Sn-1.If ρK is positive and continuous, then K will be called a star body (about the origin), and Sn denotes the set of star bodies in Rn. We will use Son and Sen to denote the subset of star bodies in Sn containing the origin in their interiors and origin-symmetric star bodies, respectively. Two star bodies K and L are said to be the dilation of one another if ρK(u)/ρL(u) is independent of u∈Sn-1.LetKn denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean Rn. For u∈Sn-1,  u⊥ denotes the (n-1)-dimensional subspace orthogonal to u. We use Vk(M) to denote the k-dimensional volume of a k-dimensional compact convex set M. Instead of Vn we usually write V. For the standard unit ball B in Rn, we write ωn=V(B) for its volume.Busemann and Petty posed a problem [3]: let K and L be origin-symmetric convex bodies in Rn. Is it true that, for any u∈Sn-1, (2)Vn-1K∩u⊥≤Vn-1L∩u⊥⟹VK≤VL?A long list of authors contributed to the solution of this famous problem over a period of 40 years; see [4–18]. The question has a negative answer for n≥5 and an affirmative answer for n=3,4. For a detailed account of the interesting history of the Busemann-Petty problem, see the books by Gardner [1, Chapter 8] and Koldobsky [19, Chapter 5].The crucial idea in the solution of the problem was to define new convex body which was called intersection body. This was done by Lutwak [15] whose work is considered the starting point of the solution of the Busemann-Petty problem in all dimensions. For K∈Son, the intersection body, IK, of K is a star body whose radial function in the direction u∈Sn-1 is equal to the (n-1)-dimensional volume of the section of K by u⊥; that is, (3)ρIK,u=Vn-1K∩u⊥.The intersection bodies have been intensively studied in recent years (see [20–28] and the books [19, 29]). From (3) and the fact that star bodies K and L satisfy K⊆L if and only if ρ(K,·)≤ρ(L,·), we see that the Busemann-Petty problem can be rephrased in the following way: for K,L∈Son, is it true that (4)IK⊆IL⟹VK≤VL?Lutwak [15] showed that the problem has an affirmative answer if the body K is restricted to the class of intersection bodies. In addition, Lutwak proved that if L is a sufficiently smooth origin-symmetric star body with positive radial function which is not an intersection body, then there exists an origin-symmetric star body K such that IK⊆IL but V(K)>V(L). Further, Busemann-Petty problems have been considered in the context of Lp Brunn-Minkowski Theory (see [30–41]). In particular, Haberl and Ludwig [42] generalized the intersection body to Lp form and introduced the notion of Lp intersection body: let L be a star body and nonzero p<1. The Lp intersection body of L,  IpL, is the symmetric star body whose radial function is defined by (5)ρIpL,up=∫Lu·x-pdx.After that, associated withLp intersection bodies, Yuan and Cheung [41] gave an affirmative form of Busemann-Petty problem for the Lp intersection bodies.Theorem 1.
LetK be Lp intersection body and L be a star body in Rn. If IpK⊆IpL, then (6)VK≤VL,for0<p<1,VK≥VL,forp<0.In both cases equality holds if and only if K=L.In 2007, Yu et al. [40] defined the quasi Lp intersection body as follows: let K be a star body and p≥1. The quasi Lp intersection body, IpK, of K is defined by (7)ρIpK,u=V~pK,B∩u⊥V~pB,B∩u⊥1/p,for u∈Sn-1, where V~p denotes the Lp dual mixed volume (see (22)).Suppose thatf is a Borel function on Sn-1. The spherical Radon transform Rf [43] of f is defined by (8)Rfu=∫Sn-1∩u⊥fvdSn-2v,for u∈Sn-1. Using the spherical Radon transform, the definition of IpK is rewritten by (9)ρIpK,u=V~pK,B∩u⊥V~pB,B∩u⊥1/p=1n-1ωn-1RρKn-pu1/p=1n-1ωn-1∫Sn-1∩u⊥ρK,vn-pdSn-2v1/p,for u∈Sn-1.One aim of this paper is to establish the volume forms of the Busemann-Petty problems for the quasiLp intersection bodies. For convenience, let Ip denote the set of quasi Lp intersection bodies.Theorem 2.
LetK,L∈Son and p≥1. If 1≤p<n and K∈Ip, then (10)IpK⊆IpL⟹VK≤VL.If p>n and L∈Ip, then (11)IpK⊆IpL⟹VK≥VL.And V(K)=V(L) if and only if K=L.Remark 3.
Theorem2 can be found in [40]. However, what should be noted is that we give a new method of proof in this paper.Theorem 4.
For1≤p<n, if K∉Sen, then there exists L∈Sen such that (12)IpK⊂IpL,but (13)VK>VL.Recall the definition ofLp dual geominimal surface area, G~p(K), of K∈Son for p>0 in [44]: (14)ωnp/nG~pK=sup⁡nV~pK,QVQ∗p/n:Q∈Sen.Another aim of this paper is to give theLp dual geominimal surface area forms of Busemann-Petty problems for the quasi Lp intersection bodies. If M∈Ip, then we rewrite the definition of Lp dual geominimal surface area by (15)ωnp/nG¯pK=sup⁡nV~pK,MVM∗p/n:M∈Ip.Theorem 5.
Ifp≥1 and K,L∈Son, then (16)IpK⊆IpL,implying (17)G¯pK≤G¯pL.And G¯p(K)=G¯p(L) if and only if K=L.Theorem 6.
For1≤p<n, if K∉Sen, then there exists L∈Sen such that (18)IpK⊂IpL,but (19)G~pK>G~pL.


## 2. Preliminaries

### 2.1.Lp Dual Mixed Volume

ForK,L∈Son,  p>0, and λ,μ≥0 (λ+μ≠0), the Lp radial combination, λ∘K+~pμ∘L∈Son, of K and L is defined by (see [40])(20)ρλ∘K+~pμ∘L,·p=λρK,·p+μρL,·p.The polar coordinate formula for the volume of a bodyK∈Son is (21)VK=1n∫Sn-1ρKundSu.Forp≥1, the Lp dual mixed volume, V~p(K,L), of K,L∈Son was defined by (see [40]) (22)npV~pK,L=limε→0+⁡VK+~pε∘L-VKε.From definition (22), the following integral representation of Lp dual mixed volume was given (see [40]): if K,L∈Son and p≥1, then (23)V~pK,L=1n∫Sn-1ρK,un-pρL,updSu.Obviously, (24)V~pK,K=VK.The Minkowski inequality for theLp dual mixed volume was established in [40]: if K,L∈Son, then, for 1≤p<n, (25)V~pK,L≤VKn-p/nVLp/n;for p>n, (26)V~pK,L≥VKn-p/nVLp/n.In every case, equality holds if and only if K is a dilation of L.


### 2.2.Lp Dual Blaschke Body

ForK,L∈Son,  0<p<n, and λ,μ≥0 (λ+μ≠0), the Lp dual Blaschke combination, λ⋆K+˘pμ⋆L∈Son, of K and L is defined by (see [28]) (27)ρλ⋆K+˘pμ⋆L,·n-p=λρK,·n-p+μρL,·n-p.Takingλ=μ=1/2,  L=-K in (27), the Lp dual Blaschke body, ∇˘pK, of K is given by (see [28]) (28)∇˘pK=12⋆K+˘p12⋆-K.Obviously, the Lp dual Blaschke body is origin-symmetric.


## 2.1.Lp Dual Mixed Volume

ForK,L∈Son,  p>0, and λ,μ≥0 (λ+μ≠0), the Lp radial combination, λ∘K+~pμ∘L∈Son, of K and L is defined by (see [40])(20)ρλ∘K+~pμ∘L,·p=λρK,·p+μρL,·p.The polar coordinate formula for the volume of a bodyK∈Son is (21)VK=1n∫Sn-1ρKundSu.Forp≥1, the Lp dual mixed volume, V~p(K,L), of K,L∈Son was defined by (see [40]) (22)npV~pK,L=limε→0+⁡VK+~pε∘L-VKε.From definition (22), the following integral representation of Lp dual mixed volume was given (see [40]): if K,L∈Son and p≥1, then (23)V~pK,L=1n∫Sn-1ρK,un-pρL,updSu.Obviously, (24)V~pK,K=VK.The Minkowski inequality for theLp dual mixed volume was established in [40]: if K,L∈Son, then, for 1≤p<n, (25)V~pK,L≤VKn-p/nVLp/n;for p>n, (26)V~pK,L≥VKn-p/nVLp/n.In every case, equality holds if and only if K is a dilation of L.


## 2.2.Lp Dual Blaschke Body

ForK,L∈Son,  0<p<n, and λ,μ≥0 (λ+μ≠0), the Lp dual Blaschke combination, λ⋆K+˘pμ⋆L∈Son, of K and L is defined by (see [28]) (27)ρλ⋆K+˘pμ⋆L,·n-p=λρK,·n-p+μρL,·n-p.Takingλ=μ=1/2,  L=-K in (27), the Lp dual Blaschke body, ∇˘pK, of K is given by (see [28]) (28)∇˘pK=12⋆K+˘p12⋆-K.Obviously, the Lp dual Blaschke body is origin-symmetric.


## 3. Proofs of Theorems2–6

The proof of Theorem2 needs the following Lemma.Lemma 7 (see [40]).
IfK,L∈Son, then, for p≥1, (29)V~pK,IpL=V~pL,IpK.Proof of Theorem2.
For a star bodyK¯ with IpK¯=K, it follows from Lemma 7 that (30)VK=V~pK,K=V~pK,IpK¯=V~pK¯,IpK;V~pL,K=V~pL,IpK¯=V~pK¯,IpL.Since (31)V~pK¯,IpK=1n∫Sn-1ρK¯,un-pρIpK,uρIpL,upρIpL,updSu≤maxu∈Sn-1⁡ρIpK,uρIpL,upV~pK¯,IpL,we have (32)VKV~pK¯,IpL≤maxu∈Sn-1⁡ρIpK,uρIpL,up.From IpK⊆IpL, we obtain that, for 1≤p<n, (33)VK≤V~pK¯,IpL=V~pL,K≤VLn-p/nVKp/n;that is,(34)VK≤VL.
From the equality condition of (25), we know that V(K)=V(L) if and only if K=L.
From a star bodyL¯ with IpL¯=L, By Lemma 7, we have (35)VL=V~pL,L=V~pL,IpL¯=V~pL¯,IpL;V~pK,L=V~pK,IpL¯=V~pL¯,IpK.Thus (36)V~pL¯,IpK=1n∫Sn-1ρL¯,un-pρIpK,uρIpL,upρIpL,updSu≤maxu∈Sn-1⁡ρIpK,uρIpL,upV~pL¯,IpL;that is, (37)V~pK,LVL≤maxu∈Sn-1⁡ρIpK,uρIpL,up. For p>n, it follows from IpK⊆IpL that (38)VL≥V~pK,L≥VKn-p/nVLp/n; namely, (39)VK≥VL.Lemma 8 (see [28]).
IfK,L∈Son and λ,μ≥0 (λ+μ≠0), then, for 0<p<n, (40)Vλ⋆K+˘pμ⋆Ln-p/n≤λVKn-p/n+μVLn-p/n,with equality if and only if K is a dilation of L.Letλ=μ=1/2,  L=-K in (40), the following is an immediate result of Lemma 8.Corollary 9.
IfK∈Son, then, for 0<p<n, (41)V∇˘pK≤VK,with equality if and only if K is origin-symmetric.Lemma 10.
IfK∈Son, then, for p≥1, (42)Ip∇˘pK=IpK.Proof.
From (9), (27), and (28), we have (43)ρIp∇˘pK,up=1n-1ωn-1∫Sn-1∩u⊥ρ12⋆K+˘p12⋆-K,vn-pdSn-2v=1n-1ωn-1∫Sn-1∩u⊥12ρK,vn-p+12ρ-K,vn-pdSn-2v=12ρIpK,up+12ρIp-K,up. Since Ip(-K)=Ip(K), we have (44)ρIp∇˘pK,up=ρIpK,up; that is, (45)Ip∇˘pK=IpK.Proof of Theorem4.
SinceK∉Sen, Corollary 9 implies (46)V∇˘pK<VK. Let ε>0 such that V((1+ε)∇˘pK)<V(K). Taking L=(1+ε)∇˘pK we have (47)VK>VL. Combining with Lemma 10 we get (48)IpL=Ip1+ε∇˘pK=1+εn-p/pIp∇˘pK=1+εn-p/pIpK⊃IpK.Proof of Theorem5.
From the conditionIpK⊆IpL, we have, for arbitrary N∈Son, (49)V~pN,IpK≤V~pN,IpL. By Lemma 7, we obtain (50)V~pK,IpN≤V~pL,IpN.Let M=IpN in (50). Together with (15), it follows that (51)G¯pK≤G¯pL,with equality if and only if K=L.Lemma 11.
IfK∈Son and 1≤p<n, then (52)G~p∇˘pK≤G~pK, with equality if and only if K is origin-symmetric.Proof.
From (14), (27), and (28), we get (53)ωnp/nG~p∇˘pK=sup⁡nV~p∇˘pK,QVQ∗p/n:Q∈Sen=sup⁡nV~p12⋆K+˘p12⋆-K,QVQ∗p/n:Q∈Sen=sup⁡∫Sn-1ρ12⋆K+˘p12⋆-K,un-pρL,updSuVQ∗p/n:Q∈Sen=sup⁡∫Sn-112ρK,un-p+12ρ-K,un-pρL,updSuVQ∗p/n:Q∈Sen=sup⁡n2V~pK,QVQ∗p/n+n2V~p-K,QVQ∗p/n:Q∈Sen≤12sup⁡nV~pK,QVQ∗p/n:Q∈Sen+12sup⁡nV~p-K,QVQ∗p/n:Q∈Sen.Note that Q∈Sen; thus we have V~p(-K,Q)=V~p(K,Q). Together with (53), this yields (54)G~p∇˘pK≤G~pK.
Equality holds in (53) if and only if K is a dilation of -K. This gives K=-K. Namely, K is origin-symmetric. Hence, equality holds in (54) if and only if K is origin-symmetric.Proof of Theorem6.
SinceK∉Sen, from Lemma 11 we have (55)G~p∇˘pK<G~pK. Choose ε>0 such that G~p((1+ε)∇˘pK)<G~p(K). Taking L=(1+ε)∇˘pK we obtain (56)G~pK>G~pL. It follows from Lemma 10 that (57)IpL=Ip1+ε∇˘pK=1+εn-p/pIp∇˘pK=1+εn-p/pIpK⊃IpK.



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