# DualL

Authors: Tongyi Ma; Yibin Feng
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

The integral formula of dualL
p-geominimal surface area is given and the concept of dual L
p-geominimal surface area is extended to dual L
p-mixed geominimal surface area. Properties for the dual L
p-mixed geominimal surface areas are established. Some inequalities, such as analogues of Alexandrov-Fenchel inequalities, Blaschke-Santaló inequalities, and affine isoperimetric inequalities for dual L
p-mixed geominimal surface areas, are also obtained.


---

## Body

## 1. Introduction

The concept of geominimal surface area was introduced by Petty [1] about 40 years ago, and its L
p-extension was first introduced by Lutwak [2, 3]. They have been proved to be key ingredients in connecting affine differential geometry, relative differential geometry, and Minkowski geometry. The basic theory concerning geominimal surface area is developed, and a close connection is established between this theory and affine differential geometry in [1]. The L
p-geominimal surface area is now thought to be at the core of the rapidly developing L
p-Brunn-Minkowski theory. Hence, it receives a lot of attention and motivates extensions of some known inequalities for geominimal surface areas to L
p-geominimal surface areas. These new inequalities of L
p-type (
p
>
1
) are stronger than their classical counterparts.However, finding an integral expression for theL
p-geominimal surface area seems to be intractable. This also leads to a big obstacle on extending the L
p-geominimal surface area. Until more recently, Zhu et al. [4] provided an integral formula for L
p-geominimal surface area by p-Petty body and introduced L
p-mixed geominimal surface areas which extended the L
p-geominimal surface area. Thereout, they established some new L
p-affine isoperimetric inequalities.Recently, Wang and Qi [5] introduced a concept of dual L
p-geominimal surface area, which is a dual concept for L
p-geominimal surface area and belongs to the dual L
p-Brunn-Minkowski theory for star bodies also developed by Lutwak (see [6, 7]). The dual L
p-Brunn-Minkowski theory for star bodies and a more extensive dual Orlicz-Brunn-Minkowski theory for star bodies received considerable attention (see, e.g., [8–21]), and they have been proved to be very powerful in solving many geometric problems, for instance, the Busemann-Petty problems (see, e.g., [6, 22–24]).In this paper, we show that the infimum in the definition of dualL
p-geominimal surface area is a minimum and provide an integral formula for dual L
p-geominimal surface area by dual p-Petty body. Moreover, we define the dual L
p-mixed geominimal surface area and establish some new L
p-affine isoperimetric inequalities for it.Our paper is organized as follows. In Section2, we provide the necessary background, such as definitions and known results which will be needed. Section 3 includes the basic theory of dual L
p-geominimal surface area, such as theorem of existence and uniqueness for dual L
p-geominimal surface area, as well as the integral definition of dual L
p-geominimal surface area. In Section 4, we introduce the dual L
p-mixed geominimal surface area and prove some important properties, such as affine invariant properties. We also obtain analogues of Alexandrov-Fenchel inequalities, Blaschke-Santaló inequalities, and affine isoperimetric inequalities for dual L
p-mixed geominimal surface areas. Finally, we investigate the dual ith L
p-mixed geominimal surface areas and obtain analogues of Blaschke-Santaló and affine isoperimetric inequalities in Section 5.


## 2. Preliminaries and Notations

LetK
n denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space R
n. For the set of convex bodies containing the origin in their interiors and the set of convex bodies whose centroids lie at the origin in R
n, we write K
o
n and K
c
n, respectively. Let V
(
K
) denote the n-dimensional volume of a body K, and let B denote the standard Euclidean unit ball in R
n and write ω
n
=
V
(
B
) for its volume, and let S
n
-
1 denote the unit sphere for B.ForK
∈
K
o
n, its support function h
K
=
h
K
,
·
:
R
n
∖
{
o
}
→
[
0
,
∞
) is defined by x
∈
R
n
∖
{
o
}, h
(
K
,
x
)
=
m
a
x
{
〈
x
,
y
〉
:
y
∈
K
}, where 〈
·
,
·
〉 is the standard inner product on R
n. Associated with each K
∈
K
o
n, one can uniquely define its polar body K
∗
∈
K
o
n by K
∗
=
{
x
∈
R
n
:
〈
x
,
y
〉
≤
1
,
∀
y
∈
K
}. It is easily verified that (
K
∗
)
∗
=
K if K
∈
K
o
n.ForK
,
L
∈
K
n and α
,
β
≥
0 (not both zero), the Minkowski linear combination α
·
K
+
β
·
L
∈
K
n is defined by (1)
h
α
·
K
+
β
·
L
,
·
=
α
h
K
,
·
+
β
h
L
,
·
.The classical Brunn-Minkowski inequality (see [25]) states that for convex bodies K
,
L
∈
K
n and real α
,
β
≥
0 (not both zero), the volume of the bodies and the volume of their Minkowski linear combination α
·
K
+
β
·
L
∈
K
n are related by(2)
V
α
·
K
+
β
·
L
1
/
n
≥
α
V
K
1
/
n
+
β
V
L
1
/
n
,with equality if and only if K and L are homothetic.For realp
≥
1, K
,
L
∈
K
o
n, and α
,
β
≥
0 (not both zero), the Firey linear combination, α
·
K
+
p
β
·
L, is defined by (see [26]) (3)
h
α
·
K
+
p
β
·
L
,
·
p
=
α
h
K
,
·
p
+
β
h
L
,
·
p
.For the Firey linear combinationα
·
K
+
p
β
·
L, Firey [26] also established the L
p-Brunn-Minkowski inequality (an inequality that is also known as the Brunn-Minkowski-Firey inequality, see [14]). If p
>
1, α
,
β
≥
0 (not both zero), and K
,
L
∈
K
o
n, then (4)
V
α
·
K
+
p
β
·
L
p
/
n
≥
α
V
K
p
/
n
+
β
V
L
p
/
n
,with equality if and only if K and L are dilates.A setK in R
n is star-shaped at o if o
∈
K and for each x
∈
R
n
∖
{
o
}, the intersection K
∩
{
c
x
:
c
≥
0
} is a (possibly degenerate) compact line segment. If K
⊂
R
n is star-shaped at the origin o, we define its radial function ρ
K for x
∈
R
n
∖
{
o
} by ρ
(
K
,
x
)
=
m
a
x
{
λ
≥
0
:
λ
x
∈
K
}. If ρ
K is positive and continuous, then K is called a star body about the origin. S
o
n denotes the set of star bodies (about the origin) in R
n. Two star bodies K and L are dilates of one another if ρ
K
(
u
)
/
ρ
L
(
u
) is independent of u
∈
S
n
-
1. Note that K
∈
S
o
n can be uniquely determined by its radial function ρ
K
(
·
) and vice versa. If α
>
0, we have(5)
ρ
K
α
x
=
α
-
1
ρ
K
x
,
ρ
α
K
x
=
α
ρ
K
x
.More generally, from the definition of the radial function, it follows immediately that for ϕ
∈
G
L
(
n
) the radial function of the image ϕ
K
=
{
ϕ
y
:
y
∈
K
} of K
∈
S
o
n is given by (see [27])(6)
ρ
ϕ
K
,
x
=
ρ
K
,
ϕ
-
1
x
,
∀
x
∈
R
n
.Obviously, for K
,
L
∈
S
o
n, (7)
K
⊆
L
i
f
f
ρ
K
≤
ρ
L
.The radial Hausdorff metric between the star bodies K and L is (8)
δ
~
K
,
L
=
m
a
x
u
∈
S
n
-
1
ρ
K
u
-
ρ
L
u
.A sequence {
K
i
} of star bodies is said to be convergent to K if (9)
δ
~
K
i
,
K
⟶
0
,
a
s
i
⟶
∞
.Therefore, a sequence of star bodies K
i converges to K if and only if the sequence of radial functions ρ
(
K
i
,
·
) converges uniformly to ρ
(
K
,
·
) (see [28, Theorem  7.9]).According to the definitions of the polar body for convex body, support function, and radial function, it follows that forK
∈
K
o
n
(10)
h
K
∗
u
ρ
K
u
=
1
,
ρ
K
∗
u
h
K
u
=
1
,
∀
u
∈
S
n
-
1
.One of the most important inequalities in convex geometry is the Blaschke-Santaló inequality about polar body (cf. [1, 27, 29]): If K
∈
K
c
n, then(11)
V
K
V
K
∗
≤
ω
n
2
,where the equality holds if and only if K is an ellipsoid.IfK
,
L
∈
S
o
n and α
,
β
≥
0 (not both zero), then, for p
≥
1, the radial harmonic L
p-combination, α
⋄
K
+
^
-
p
β
⋄
L
∈
S
o
n, is defined by (see [3]) (12)
ρ
α
⋄
K
+
^
-
p
β
⋄
L
,
·
-
p
=
α
ρ
K
,
·
-
p
+
β
ρ
L
,
·
-
p
.Forp
≥
1 and K
,
L
∈
S
o
n, the dual harmonic L
p-mixed volume, V
~
-
p
(
K
,
L
), is defined by (13)
-
n
p
V
~
-
p
K
,
L
=
l
i
m
ε
→
0
+
V
K
+
^
-
p
ε
⋄
L
-
V
K
ε
.Let K
,
L
∈
S
o
n and p
≥
1. Then, the integral representation of dual harmonic L
p-mixed volume of K and L, V
~
-
p
(
K
,
L
), is given (see [3]):(14)
V
~
-
p
K
,
L
=
1
n
∫
S
n
-
1
ρ
K
n
+
p
u
ρ
L
-
p
u
d
S
u
.With (5) and (14) taken together, we obtain for α
>
0
(15)
V
~
-
p
α
K
,
L
=
α
n
+
p
V
~
-
p
K
,
L
,
V
~
-
p
K
,
α
L
=
α
-
p
V
~
-
p
K
,
L
.In [3], Lutwak proved the following: For K
,
L
∈
S
o
n and α
,
β
≥
0, if p
≥
1, then, for ϕ
∈
G
L
(
n
), (16)
ϕ
α
⋄
K
+
^
p
β
⋄
L
=
α
⋄
ϕ
K
+
^
p
β
⋄
ϕ
L
.SinceV
(
ϕ
K
)
=
d
e
t
ϕ
V
(
K
), for all K
∈
S
o
n and ϕ
∈
G
L
(
n
), the following follows from (6), (14), and (15).Proposition 1.
Ifp
≥
1 and K
,
L
∈
S
o
n, then, for ϕ
∈
G
L
(
n
), (17)
V
~
-
p
ϕ
K
,
ϕ
L
=
det
⁡
ϕ
V
~
-
p
K
,
L
.The caseϕ
∈
S
L
(
n
) of Proposition 1 reduces to the following formula:(18)
V
~
-
p
ϕ
K
,
ϕ
L
=
V
~
-
p
K
,
L
.This integral representation ofV
~
-
p
(
·
,
·
), with Hölder’s inequality (see [30, p. 140]) together with the polar coordinate formula, immediately gives the following:(19)
V
~
-
p
K
,
L
n
≥
V
K
n
+
p
V
L
-
p
,with equality if and only if K and L are dilates.The following result is an immediate consequence of (19).Lemma 2.
Suppose thatp
≥
1 and U
⊂
S
o
n such that K
,
L
∈
U. If for all Q
∈
U
(20)
V
~
-
p
K
,
Q
=
V
~
-
p
L
,
Q
or
V
~
-
p
Q
,
K
=
V
~
-
p
Q
,
L
,then K
=
L.The continuity of the dual harmonicL
p-mixed volume V
~
-
p
:
S
o
n
×
S
o
n
→
(
0
,
∞
) is contained.Lemma 3.
Suppose that sequences{
K
i
}
,
{
L
j
}
⊂
S
o
n and K
i
→
K
∈
S
o
n, L
j
→
L
∈
S
o
n. If p
≥
1, then l
i
m
i
,
j
→
∞
V
~
-
p
(
K
i
,
L
j
)
=
V
~
-
p
(
K
,
L
).Proof.
SinceK
i
→
K and L
j
→
L are equivalent to ρ
K
i
→
ρ
K and ρ
L
j
→
ρ
L, uniformly on S
n
-
1, and ρ
K, ρ
L are positively continuous on S
n
-
1, then ρ
K
i and ρ
L
j are uniformly bounded on S
n
-
1 (see [28, Theorem  7.9]). Hence, (21)
ρ
K
i
n
+
p
⟶
ρ
K
n
+
p
,
u
n
i
f
o
r
m
l
y
o
n
S
n
-
1
,
ρ
L
j
-
p
⟶
ρ
L
-
p
,
u
n
i
f
o
r
m
l
y
o
n
S
n
-
1
.Hence, (22)
∫
S
n
-
1
ρ
K
i
n
+
p
u
ρ
L
j
-
p
u
d
S
u
⟶
∫
S
n
-
1
ρ
K
n
+
p
u
ρ
L
-
p
u
d
S
u
,
i
f
i
,
j
⟶
∞
.Namely, l
i
m
i
,
j
→
∞
V
~
-
p
(
K
i
,
L
j
)
=
V
~
-
p
(
K
,
L
).The volume-normalized dual conical measured
V
~
K
∗ of K
∈
S
o
n is defined by V
(
K
)
d
V
~
K
∗
=
1
/
n
ρ
K
n
d
S, where S is Lebesgue measure on S
n
-
1. We shall make use of the fact that the volume-normalized dual conical measure V
~
K
∗ is a probability measure on S
n
-
1.The following lemma will be needed.Lemma 4 (see [3]).
LetC
n denote the set of compact convex subsets of Euclidean n-space R
n, and suppose K
i
∈
K
o
n such that K
i
→
L
∈
C
n. If the sequence V
(
K
i
∗
) is bounded, then L
∈
K
o
n.


## 3. The DualL
p-Geominimal Surface Area

Based on the notion of dualL
p-mixed volumes, Wang and Qi [5] defined the dual L
p-geominimal surface area as follows: For K
∈
S
o
n, the dual L
p-geominimal surface area, G
~
-
p
(
K
), of K is defined by(23)
ω
n
-
p
/
n
G
~
-
p
K
=
inf
⁡
n
V
~
-
p
K
,
Q
V
Q
∗
-
p
/
n
:
Q
∈
K
o
n
.For this notion ofL
p-dual geominimal surface area, Wang and Qi in [5] established the following affine isoperimetric inequality and Blaschke-Santaló type inequality: For K
∈
S
o
n and p
≥
1,(24)
G
~
-
p
K
≥
n
ω
n
-
p
/
n
V
K
n
+
p
/
n
,with equality if and only if K is an ellipsoid centred at the origin.IfK
∈
K
c
n and p
≥
1, then(25)
G
~
-
p
K
G
~
-
p
K
∗
≤
n
ω
n
2
,with equality if and only if K is an ellipsoid.By the homogeneity of volume and dualL
p-mixed volume, the dual L
p-geominimal surface area could also be defined by (26)
G
~
-
p
K
=
inf
⁡
n
V
~
-
p
K
,
Q
:
Q
∈
K
o
n
,
V
Q
∗
=
ω
n
.It will be shown that the infimum in the above definition is attained.Theorem 5.
IfK
∈
S
o
n and p
≥
1, then there exists a unique body K
~
∈
K
o
n such that (27)
G
~
-
p
K
=
n
V
~
-
p
K
,
K
~
,
V
K
~
∗
=
ω
n
.Proof.
From the definition ofG
~
-
p
(
K
), there exists a sequence {
M
i
}
⊂
K
o
n such that V
(
M
i
∗
)
=
ω
n, with V
~
-
p
(
K
,
B
)
≥
V
~
-
p
(
K
,
M
i
), for all i, and n
V
~
-
p
(
K
,
M
i
)
→
G
~
-
p
(
K
). To see that the M
i
∈
K
o
n
,
i
=
1,2
,
…, are uniformly bounded, let (28)
R
i
=
R
M
i
=
ρ
M
i
,
u
i
=
max
⁡
ρ
M
i
,
u
:
u
∈
S
n
-
1
,where u
i is any of the points in S
n
-
1 at which this maximum is attained. Let r
K
=
m
i
n
S
n
-
1
ρ
K. Then, r
K
B
⊆
K. From definition (14) of dual harmonic L
p-mixed volume and Jensen’s inequality, it follows that(29)
V
~
-
p
K
,
B
V
K
≥
V
~
-
p
K
,
M
i
V
K
=
∫
S
n
-
1
ρ
K
u
ρ
M
i
u
p
d
V
~
K
∗
≥
∫
S
n
-
1
ρ
K
u
ρ
M
i
u
d
V
~
K
∗
p
≥
∫
S
n
-
1
ρ
K
u
R
i
d
V
~
K
∗
p
=
1
n
V
K
R
i
∫
S
n
-
1
ρ
K
u
n
+
1
d
S
u
p
≥
r
K
n
V
K
R
i
∫
S
n
-
1
ρ
K
u
n
d
S
u
p
=
r
K
R
i
p
.Namely, (30)
ω
n
r
K
n
r
K
R
i
p
≤
V
~
-
p
K
,
M
i
≤
V
~
-
p
K
,
B
<
∞for a fixed K
∈
S
o
n; then, the sequence {
M
i
} is uniformly bounded.
Since the sequence{
M
i
} is uniformly bounded, the Blaschke selection theorem guarantees the existence of a subsequence of M
i, which will also be denoted by M
i, and a compact convex L
∈
C
n, such that M
i
→
L. Since V
(
M
i
∗
)
=
ω
n, Lemma 4 gives L
∈
K
o
n. Now, M
i
→
L implies that M
i
∗
→
L
∗, and since V
(
M
i
∗
)
=
ω
n, it follows that V
(
L
∗
)
=
ω
n. Lemma 3 can now be used to conclude that L will serve as the desired body K
~.
The uniqueness of the minimizing body is easily demonstrated as follows. SupposeL
1
,
L
2
∈
K
o
n and L
1
≠
L
2, such that V
(
L
1
∗
)
=
ω
n
=
V
(
L
2
∗
), and (31)
V
~
-
p
K
,
L
1
=
V
~
-
p
K
,
L
2
.
DefineL
∈
K
o
n by (32)
L
=
1
2
⋄
L
1
+
^
-
1
1
2
⋄
L
2
.
Since, obviously,(33)
L
∗
=
1
2
·
L
1
∗
+
1
2
·
L
2
∗and V
(
L
1
∗
)
=
ω
n
=
V
(
L
2
∗
), it follows from Brunn-Minkowski inequality (2) that (34)
V
L
∗
≥
ω
n
,with equality if and only if L
1
=
L
2.
By formula (14) of dual L
p-mixed volume, together with the convexity of ϕ
(
t
)
=
t
p
(
p
≥
1
), we have(35)
V
~
-
p
K
,
L
=
1
n
∫
S
n
-
1
ρ
K
u
ρ
1
/
2
∘
L
1
+
~
-
1
1
/
2
∘
L
2
u
p
ρ
K
u
n
d
S
u
=
1
n
∫
S
n
-
1
ρ
K
u
1
/
2
ρ
L
1
u
+
1
/
2
ρ
L
2
u
-
1
p
ρ
K
u
n
d
S
u
=
1
n
∫
S
n
-
1
ρ
K
u
2
ρ
L
1
u
+
ρ
K
u
2
ρ
L
2
u
p
ρ
K
u
n
d
S
u
≤
1
2
n
∫
S
n
-
1
ρ
K
u
ρ
L
1
u
p
ρ
K
u
n
d
S
u
+
1
2
n
∫
S
n
-
1
ρ
K
u
ρ
L
2
u
p
ρ
K
u
n
d
S
u
=
1
2
V
~
-
p
K
,
L
1
+
1
2
V
~
-
p
K
,
L
2
=
V
~
-
p
K
,
L
1
=
V
~
-
p
K
,
L
2
,with equality if and only if L
1
=
L
2. Thus, (36)
V
~
-
p
K
,
L
V
L
∗
-
p
/
n
<
V
~
-
p
K
,
L
1
V
L
1
∗
-
p
/
n
=
V
~
-
p
K
,
L
2
V
L
2
∗
-
p
/
nis the contradiction that would arise if it were the case that L
1
≠
L
2. This completes the proof.The unique body whose existence is guaranteed by Theorem5 will be denoted by T
~
p
K and will be called the dual p-Petty body of K. The polar body of T
~
p
K will be denoted by T
~
p
∗
K rather than (
T
~
p
K
)
∗. Thus, for K
∈
S
o
n, the body T
~
-
p
K is defined by (37)
G
~
-
p
K
=
n
V
~
-
p
K
,
T
~
p
K
,
V
T
~
p
∗
K
=
ω
n
.ForK
∈
K
n, there exists a unique point s
(
K
) in the interior of K, called the Santaló point of K, such that (see [3]) (38)
V
-
s
K
+
K
∗
=
m
i
n
V
-
x
+
K
∗
:
x
∈
int
⁡
K
,or, for the unique s
(
K
)
∈
K, this is equivalent to (39)
∫
S
n
-
1
u
h
-
s
K
+
K
,
u
-
n
+
1
d
S
u
=
0
.LetK
s
n denote the set of convex bodies having their Santaló point at the origin in R
n. Thus, we have (see [3]) (40)
K
∈
K
s
n
i
f
f
K
∗
∈
K
c
n
.Let (41)
T
n
=
T
~
∈
K
n
:
s
K
=
o
,
V
T
~
∗
=
ω
n
.The next result is an immediate consequence of Theorem5.Theorem 6.
For eachK
∈
S
o
n, there exists a unique body T
~
p
K
∈
T
n with G
~
-
p
(
K
)
=
n
V
~
-
p
(
K
,
T
~
p
K
).The unique bodyT
~
p
K is called the dual p-Petty body of K.By Theorem6 and the integral representation (14) of dual harmonic L
p-mixed volume, we have the following integral formula of G
~
-
p
(
K
).Proposition 7.
For eachK
∈
S
o
n, there exists a unique convex body T
~
=
T
~
p
K
∈
T
n with (42)
G
~
-
p
K
=
∫
S
n
-
1
ρ
K
n
+
p
p
ρ
T
~
-
p
u
d
S
u
.


## 4. The DualL
p-Mixed Geominimal Surface Area

Motivated by the definition ofL
p-mixed geominimal surface area of Zhu et al. (see [4]), we now define the dual L
p-mixed geominimal surface area as follow: For each K
i
∈
S
o
n, i
=
1
,
…
,
n, and p
≥
1, there exists a unique convex body (dual p-Petty body of K
i) T
~
i
=
T
~
p
K
i
∈
T
n
(
i
=
1
,
…
,
n
) with (43)
G
~
-
p
K
1
,
…
,
K
n
=
∫
S
n
-
1
ρ
K
1
n
+
p
p
ρ
T
~
1
-
p
u
⋯
ρ
K
n
n
+
p
u
ρ
T
~
n
-
p
u
1
/
n
d
S
u
.
G
~
-
p
(
K
1
,
…
,
K
n
) will be called the dual L
p-mixed geominimal surface area of K
1
,
…
,
K
n
∈
S
o
n.Letg
-
p
(
K
i
,
u
)
=
ρ
K
i
n
+
p
(
p
)
ρ
T
~
i
-
p
(
u
). Then, G
~
-
p
(
K
1
,
…
,
K
n
) can be written as follows:(44)
G
~
-
p
K
1
,
…
,
K
n
=
∫
S
n
-
1
g
-
p
K
1
,
u
⋯
g
-
p
K
n
,
u
1
/
n
d
S
u
.The following propositions will provide that the dualL
p-mixed geominimal surface area is affine invariant.Proposition 8.
IfK
∈
S
o
n and every ϕ
∈
G
L
(
n
), then (45)
G
~
-
p
ϕ
K
=
det
⁡
ϕ
n
+
p
/
n
G
~
-
p
K
.Proof.
From definition (23) of dual L
p-geominimal surface area and (18), for ϕ
∈
S
L
(
n
), we have(46)
G
~
-
p
ϕ
K
=
inf
⁡
n
V
~
-
p
ϕ
K
,
Q
:
Q
∈
K
o
n
,
V
Q
∗
=
ω
n
=
inf
⁡
n
V
~
-
p
K
,
ϕ
-
1
Q
:
ϕ
-
1
Q
∈
K
o
n
,
V
ϕ
-
1
Q
∗
=
V
ϕ
t
Q
∗
=
ω
n
=
G
~
ϕ
K
.
On the other hand, forλ
>
0, it follows from (23) that (47)
G
~
-
p
λ
K
=
λ
n
+
p
G
~
-
p
K
.
Therefore, for everyϕ
∈
G
L
(
n
), we have (48)
G
~
-
p
ϕ
K
=
det
⁡
ϕ
n
+
p
/
n
G
~
-
p
K
.Proposition 9.
IfK
∈
S
o
n, then, for ϕ
∈
G
L
(
n
), (49)
det
⁡
ϕ
1
/
n
T
~
p
ϕ
K
=
ϕ
T
~
p
K
.Proof.
From the definition ofT
~
p and Proposition 8, it follows that (50)
n
V
~
-
p
K
,
T
~
p
K
=
G
~
-
p
K
=
det
⁡
ϕ
-
n
+
p
/
n
G
~
-
p
ϕ
K
=
det
⁡
ϕ
-
n
+
p
/
n
n
V
~
-
p
ϕ
K
,
T
~
p
ϕ
K
.
From the definition ofT
~
p, Proposition 9, and (15),(51)
V
~
-
p
K
,
T
~
p
K
=
det
⁡
ϕ
-
n
+
p
/
n
V
~
-
p
ϕ
K
,
T
~
p
ϕ
K
=
det
⁡
ϕ
-
1
V
~
-
p
ϕ
K
,
det
⁡
ϕ
1
/
n
T
~
p
ϕ
K
=
V
~
-
p
K
,
ϕ
-
1
det
⁡
ϕ
1
/
n
T
~
p
ϕ
K
.Namely, from Lemma 2, for each ϕ
∈
G
L
(
n
), (52)
det
⁡
ϕ
1
/
n
T
~
p
ϕ
K
=
ϕ
T
~
p
K
.Proposition 10.
Ifp
≥
1 and K
1
,
…
,
K
n
∈
S
o
n, then, for ϕ
∈
G
L
(
n
), (53)
G
~
-
p
ϕ
K
1
,
…
,
ϕ
K
n
=
det
⁡
ϕ
n
+
p
/
n
G
~
-
p
K
1
,
…
,
K
n
.In particular, if ϕ
∈
S
L
(
n
), then G
~
-
p
(
K
1
,
…
,
K
n
) is affine invariant; that is, (54)
G
~
-
p
ϕ
K
1
,
…
,
ϕ
K
n
=
G
~
-
p
K
1
,
…
,
K
n
.Proof.
SinceK
∈
S
o
n, for ϕ
∈
G
L
(
n
) and any u
∈
S
n
-
1, we have(55)
g
-
p
ϕ
K
,
u
=
ρ
ϕ
K
,
u
n
+
p
ρ
T
~
p
ϕ
K
,
u
-
p
=
ρ
ϕ
K
,
u
n
+
p
ρ
det
⁡
ϕ
-
1
/
n
ϕ
T
~
p
K
,
u
-
p
=
det
⁡
ϕ
p
/
n
ϕ
-
1
u
-
n
ρ
K
v
n
+
p
ρ
T
~
p
K
v
-
p
=
det
⁡
ϕ
p
/
n
ϕ
-
1
u
-
n
g
-
p
K
,
v
,where v
=
(
ϕ
-
1
u
)
/
ϕ
-
1
u
∈
S
n
-
1. Therefore, for ϕ
∈
G
L
(
n
), we have(56)
G
~
-
p
ϕ
K
1
,
…
,
ϕ
K
n
=
det
⁡
ϕ
p
/
n
ϕ
-
1
u
-
n
∫
S
n
-
1
g
-
p
K
1
,
u
⋯
g
-
p
K
n
,
u
1
/
n
d
S
ϕ
ϕ
-
1
u
·
ϕ
-
1
u
ϕ
-
1
u
=
det
⁡
ϕ
n
+
p
/
n
G
~
-
p
K
1
,
…
,
K
n
.The dual mixed volumeV
~
(
K
1
,
…
,
K
n
) of sets K
1
,
…
,
K
n
∈
S
o
n is defined by (57)
V
~
K
1
,
…
,
K
n
=
1
n
∫
S
n
-
1
ρ
K
1
u
⋯
ρ
K
n
u
d
S
u
.The classical dual Alexandrov-Fenchel inequalities for dual mixed volumes (cf. [27, 31, 32]) can be written as (58)
V
~
K
1
,
…
,
K
n
m
≤
∏
i
=
0
m
-
1
V
~
K
1
,
…
,
K
n
-
m
,
K
n
-
i
,
…
,
K
n
-
i
︸
m
,with equality if K
n
-
m
+
1
,
…
,
K
n are dilates of each other. If m
=
1, equality holds trivially.In particular, takingm
=
n in the above inequality and noticing that V
~
(
K
)
=
V
(
K
), we have(59)
V
~
K
1
,
…
,
K
n
n
≤
V
K
1
⋯
V
K
n
,with equality if and only if K
1
,
…
,
K
n are dilates.TakeK
1
=
⋯
=
K
n
-
i
=
K
,
K
n
-
i
+
1
=
⋯
=
K
n
=
B in V
~
(
K
1
,
…
,
K
n
), and (60)
V
~
K
,
…
,
K
︸
n
-
i
,
B
,
…
,
B
︸
i
≔
W
~
i
K
,where W
~
i
(
K
) is called the ith dual quermassintegral of K
∈
S
o
n.The following inequalities are the analogous of dual Alexandrov-Fenchel inequalities for dualL
p-mixed geominimal surface area.Theorem 11.
Ifp
≥
1 and K
1
,
…
,
K
n
∈
S
o
n, then, for 1
≤
m
≤
n, (61)
G
~
-
p
K
1
,
…
,
K
n
m
≤
∏
i
=
0
m
-
1
G
~
-
p
K
1
,
…
,
K
n
-
m
,
K
n
-
i
,
…
,
K
n
-
i
︸
m
,with equality if K
n
-
m
+
1
,
…
,
K
n are dilates of each other. If m
=
1, equality holds trivially.
In particular, ifm
=
n in the above inequality, then(62)
G
~
-
p
K
1
,
…
,
K
n
n
≤
G
~
-
p
K
1
⋯
G
~
-
p
K
n
,with equality if K
i
(
1
≤
i
≤
n
) are dilates of each other.Proof.
LetY
0
(
u
)
=
[
g
-
p
(
K
1
,
u
)
⋯
g
-
p
(
K
n
-
m
,
u
)
]
1
/
n and Y
i
+
1
(
u
)
=
[
g
-
p
(
K
n
-
i
,
u
)
]
1
/
n for i
=
0
,
…
,
m
-
1. By Hölder’s inequality (cf. [30]), we have(63)
G
~
-
p
K
1
,
…
,
K
n
=
∫
S
n
-
1
g
-
p
K
1
,
u
⋯
g
-
p
K
n
,
u
1
/
n
d
S
u
=
∫
S
n
-
1
Y
0
u
Y
1
u
⋯
Y
m
u
d
S
u
≤
∏
i
=
0
m
-
1
Y
0
u
Y
i
+
1
u
m
d
S
u
1
/
m
=
∏
i
=
0
m
-
1
G
~
-
p
1
/
m
K
1
,
…
,
K
n
-
m
,
K
n
-
i
,
…
,
K
n
-
i
︸
m
.The equality in Hölder’s inequality holds if and only if Y
0
(
u
)
Y
i
+
1
m
=
λ
i
j
m
Y
0
(
u
)
Y
j
+
1
m for some λ
i
j
>
0 and all 0
≤
i
≠
j
≤
m
-
1. This is equivalent to ρ
K
n
-
i
n
+
p
(
u
)
ρ
T
~
n
-
i
-
p
(
u
)
=
λ
i
j
n
ρ
K
n
-
j
n
+
p
(
u
)
ρ
T
~
n
-
j
-
p
(
u
). From Proposition 9, T
~
p
K
=
T
~
p
c
K for constant c
>
0. Thus, the equality holds if K
n
-
i and K
n
-
j are dilates of each other.A lemma of the following type will be needed.Lemma 12.
IfK
∈
K
c
n and p
≥
1, then(64)
G
~
-
p
K
≤
n
ω
n
2
n
-
p
/
n
V
K
∗
-
n
+
p
/
n
,with equality if and only if K is a ball centred at the origin.Proof.
From definition (23) and inequality (19), we have(65)
ω
n
-
p
/
n
V
K
∗
n
+
p
/
n
G
~
-
p
K
=
inf
⁡
n
V
~
-
p
K
,
Q
V
K
∗
n
+
p
/
n
V
Q
∗
-
p
/
n
:
Q
∈
K
c
n
≤
inf
⁡
n
V
~
-
p
K
,
Q
V
~
-
p
K
∗
,
Q
∗
:
Q
∈
K
o
n
.Since K
∈
K
o
n, taking Q
=
K, it follows from inequalities (65) and (11) that (66)
ω
n
-
p
/
n
V
K
∗
n
+
p
/
n
G
~
-
p
K
≤
inf
⁡
n
V
K
V
K
∗
:
K
∈
K
o
n
≤
n
ω
n
2
.Namely, (67)
G
~
-
p
K
≤
n
ω
n
2
n
-
p
/
n
V
K
∗
-
n
+
p
/
n
.By the equality condition of (19) and (65), we see that equality holds in (64) if and only if K is a ball centred at the origin.Now, we prove the affine isoperimetric inequalities for dualL
p-mixed geominimal surface areas.Theorem 13.
LetK
1
,
…
,
K
n
∈
K
c
n and p
≥
1; then,(68)
G
~
-
p
K
1
,
…
,
K
n
G
~
-
p
B
,
…
,
B
≤
V
~
K
1
∗
,
…
,
K
n
∗
V
~
B
,
…
,
B
-
n
+
p
/
n
,with equality if and only if K
1
,
…
,
K
n are balls centred at the origin that are dilates of each other.Proof.
From (77) in Section 5, it follows that T
~
p
B
=
B. Then, g
-
p
(
B
,
u
)
=
1, G
~
-
p
(
B
)
=
n
ω
n, and G
~
-
p
(
B
,
…
,
B
)
=
n
ω
n. By inequalities (62), (64), and (59), we have(69)
G
~
-
p
K
1
,
…
,
K
n
G
~
-
p
B
,
…
,
B
≤
V
K
1
∗
V
B
⋯
V
K
n
∗
V
B
-
n
+
p
/
n
≤
V
~
K
1
∗
,
…
,
K
n
∗
V
~
B
,
…
,
B
-
n
+
p
/
n
.By the equality condition of (62), (64), and (59), we see that equality holds in (68) if and only if K is a ball centred at the origin.Corollary 14.
LetK
1
,
…
,
K
n
∈
K
c
n and p
≥
1; then,(70)
G
~
-
p
K
1
,
…
,
K
n
≤
n
ω
n
2
n
+
p
/
n
V
~
K
1
∗
,
…
,
K
n
∗
-
n
+
p
/
n
,with equality if and only if K
1
,
…
,
K
n are balls centred at the origin that are dilates of each other.TakeK
1
=
⋯
=
K
n
-
i
=
K
,
K
n
-
i
+
1
=
⋯
=
K
n
=
B in (70), and we write (71)
G
~
-
p
K
,
…
,
K
︸
n
-
i
,
B
,
…
,
B
︸
i
≔
G
~
-
p
,
i
K
.Corollary 15.
LetK
∈
K
c
n and p
≥
1; then, for i
=
0,1
,
…
,
n
-
1, (72)
G
~
-
p
,
i
K
≤
n
ω
n
2
n
+
p
/
n
W
~
i
K
∗
-
n
+
p
/
n
,with equality if and only if K is a ball centred at the origin.


## 5. The Dualith L
p-Mixed Geominimal Surface Area

This section is mainly dedicated to investigating the dualith L
p-mixed geominimal surface area.ForK
,
L
∈
S
o
n, p
≥
1, and i
∈
R, we define dual ith L
p-mixed geominimal surface area, G
~
-
p
(
K
,
L
), of K, L as(73)
G
~
-
p
,
i
K
,
L
=
∫
S
n
-
1
g
-
p
K
,
u
n
-
i
/
n
g
-
p
L
,
u
i
/
n
d
S
uand write(74)
G
~
-
p
,
i
K
,
B
=
G
~
-
p
,
i
K
.By Theorem6, we have (75)
G
~
-
p
B
=
n
V
~
-
p
B
,
T
~
p
B
,and, obviously, (76)
G
~
-
p
B
=
n
ω
n
=
n
V
~
-
p
B
,
B
.Thus, the above two equations and the uniqueness part of Theorem 6 show that(77)
T
~
p
B
=
B
.Noticing thatg
-
p
(
B
,
u
)
=
1 for u
∈
S
n
-
1, then (78)
G
~
-
p
,
i
K
=
∫
S
n
-
1
g
-
p
K
,
u
n
-
i
/
n
d
S
u
.By (44), (73), and (74), we have(79)
G
~
-
p
,
0
K
=
G
~
-
p
K
,
G
~
-
p
,
i
K
,
K
=
G
~
-
p
K
,
(80)
G
~
-
p
,
0
K
,
L
=
G
~
-
p
K
,
G
~
-
p
,
n
K
,
L
=
G
~
-
p
L
.The following cyclic inequality for the dualith L
p-mixed geominimal surface area will be established.Theorem 16.
ForK
,
L
∈
S
o
n, p
≥
1, i
,
j
,
k
∈
R, and i
<
j
<
k, we have(81)
G
~
-
p
,
i
K
,
L
k
-
j
G
~
-
p
,
k
K
,
L
j
-
i
≥
G
~
-
p
,
j
K
,
L
k
-
i
,with equality if K and L are dilates of each other.Proof.
From definition (73) and Hölder’s inequality, it follows that, for p
≥
1,(82)
G
~
-
p
,
i
K
,
L
k
-
j
/
k
-
i
G
~
-
p
,
k
K
,
L
j
-
i
/
k
-
i
=
∫
S
n
-
1
g
-
p
K
,
u
n
-
i
/
n
g
-
p
L
,
u
i
/
n
d
S
u
k
-
j
/
k
-
i
∫
S
n
-
1
g
-
p
K
,
u
n
-
k
/
n
g
-
p
L
,
u
k
/
n
d
S
u
j
-
i
/
k
-
i
=
∫
S
n
-
1
g
-
p
K
,
u
n
-
i
k
-
j
/
n
k
-
i
g
-
p
L
,
u
i
k
-
j
/
n
k
-
i
k
-
i
/
k
-
j
d
S
u
k
-
j
/
k
-
i
∫
S
n
-
1
g
-
p
K
,
u
n
-
k
j
-
i
/
n
k
-
i
g
-
p
L
,
u
k
j
-
i
/
n
k
-
i
k
-
i
/
j
-
i
d
S
u
j
-
i
/
k
-
i
≥
∫
S
n
-
1
g
-
p
K
,
u
n
-
j
/
n
k
-
i
g
-
p
L
,
u
j
/
n
d
S
u
=
G
~
-
p
,
j
K
,
L
.That is, (83)
G
~
-
p
,
i
K
,
L
k
-
j
G
~
-
p
,
k
K
,
L
j
-
i
≥
G
~
-
p
,
j
K
,
L
k
-
i
.We obtain inequality (81). By the condition of equality in Hölder’s inequality, the equality holds in (81) if and only if, for any u
∈
S
n
-
1, (84)
g
-
p
K
,
u
n
-
i
/
n
g
-
p
L
,
u
i
/
n
g
-
p
K
,
u
n
-
k
/
n
g
-
p
L
,
u
k
/
nis a constant; that is, g
-
p
(
K
,
u
)
/
g
-
p
(
L
,
u
) is a constant for any u
∈
S
n
-
1. By the same argument in the proof of Theorem 11, we conclude that equality holds if K and L are dilates of each other.TakingL
=
B in Theorem 16 and using (74), we immediately obtain the following.Corollary 17.
ForK
∈
S
o
n, p
≥
1, i
,
j
,
k
∈
R, and i
<
j
<
k, then (85)
G
~
-
p
,
i
K
k
-
j
G
~
-
p
,
k
K
j
-
i
≥
G
~
-
p
,
j
K
k
-
i
,with equality if K is a ball centered at the origin.Then, the following Minkowski inequality for the dualith L
p-mixed geominimal surface area will be obtained.Theorem 18.
ForK
,
L
∈
S
o
n, p
≥
1, and i
∈
R and then for i
<
0 or i
>
n,(86)
G
~
-
p
,
i
K
,
L
n
≥
G
~
-
p
K
n
-
i
G
~
-
p
L
i
,and for 0
<
i
<
n,(87)
G
~
-
p
,
i
K
,
L
n
≤
G
~
-
p
K
n
-
i
G
~
-
p
L
i
.Each inequality holds as an equality if K and L are dilates of each other. For i
=
0 or i
=
n, (86) (or (87)) is identical.Proof.
(i) Fori
<
0, let (
i
,
j
,
k
)
=
(
i
,
0
,
n
) in Theorem 16; we obtain (88)
G
~
-
p
,
i
K
,
L
n
G
~
-
p
,
n
K
,
L
-
i
≥
G
~
-
p
,
0
K
,
L
n
-
i
,with equality if K and L are dilates of each other. From (80), we can get (89)
G
~
-
p
,
i
K
,
L
n
≥
G
~
-
p
K
n
-
i
G
~
-
p
L
i
,with equality if K and L are dilates of each other.
(ii) Fori
>
n, let (
i
,
j
,
k
)
=
(
0
,
n
,
i
) in Theorem 16; we obtain (90)
G
~
-
p
,
0
K
,
L
i
-
n
G
~
-
p
,
i
K
,
L
n
≥
G
~
-
p
,
n
K
,
L
i
,with equality if K and L are dilates of each other.
From (80), we can also get inequality (86).
(iii) For0
<
i
<
n, let (
i
,
j
,
k
)
=
(
0
,
i
,
n
) in Theorem 16; we obtain (91)
G
~
-
p
,
0
K
,
L
n
-
i
G
~
-
p
,
n
K
,
L
i
≥
G
~
-
p
,
i
K
,
L
n
,with equality if K and L are dilates of each other.
From (80), we can get inequality (87).
(iv) Fori
=
0 (or i
=
n), by (80), one can see (86) (or (87)) is identical.LetL
=
B in Theorem 18, G
~
-
p
(
B
)
=
n
ω
n, and (74) will lead to the following.Corollary 19.
ForK
∈
S
o
n, p
≥
1, and i
∈
R and then for i
<
0 or i
>
n,(92)
G
~
-
p
,
i
K
n
≥
n
ω
n
i
G
~
-
p
K
n
-
i
,and for 0
<
i
<
n,(93)
G
~
-
p
,
i
K
n
≤
n
ω
n
i
G
~
-
p
K
n
-
i
.Each inequality holds as an equality if K is a ball centered at the origin. For i
=
0 or i
=
n, (92) (or (93)) is identical.Now we will give an extended form of inequality (24) as follows.Theorem 20.
IfK
∈
S
o
n, p
≥
1, i
∈
R, and i
≤
0, then(94)
G
~
-
p
,
i
K
≥
n
ω
n
n
-
p
i
-
n
p
/
n
2
V
K
n
+
p
n
-
i
/
n
2
,with equality if and only if K is an ellipsoid centred at the origin.Proof.
By inequalities (92) and (24), we can immediately obtain inequality (94).As the extension of inequality (25), we obtain an analogue of Blaschke-Santaló inequality for the dual ith L
p-mixed geominimal surface area.Theorem 21.
IfK
,
L
∈
S
o
n, p
≥
1, i
∈
R, and 0
≤
i
≤
n, then (95)
G
~
-
p
,
i
K
,
L
G
~
-
p
,
i
K
∗
,
L
∗
≤
n
ω
n
2
,and equality holds for 0
<
i
<
n if K and L are dilated ellipsoids of each other centered at the origin. The inequality holds as an equality for i
=
0 (or i
=
n) if K (or L) is an ellipsoid centered at the origin.Proof.
Given (87) together with (25), it follows that(96)
G
~
-
p
,
i
K
,
L
G
~
-
p
,
i
K
∗
,
L
∗
≤
G
~
-
p
K
G
~
-
p
K
∗
n
-
i
/
n
G
~
-
p
L
G
~
-
p
L
∗
i
/
n
≤
n
ω
n
2
n
-
i
/
n
n
ω
n
2
i
/
n
=
n
ω
n
2
.The equality holds for 0
<
i
<
n if K and L are dilated ellipsoids of each other. The inequality holds as an equality for i
=
0 (or i
=
n) if K (or L) is an ellipsoid.Recall Ye’s isoperimetric inequality (see [33]): If K
∈
S
o
n, p
∈
(
0
,
∞
)
∪
(
-
∞
,
-
n
), and the dual L
p-surface area S
~
p
(
K
)
=
n
V
~
-
p
(
K
,
B
), then (97)
S
~
p
K
S
~
p
B
≥
V
K
V
B
n
+
p
/
n
.On the other hand, if p
∈
(
-
n
,
0
), then (98)
S
~
p
K
S
~
p
B
≤
V
K
V
B
n
+
p
/
n
.The equality in every inequality holds if and only if K is an origin-symmetric Euclidean ball.We now establish generalized isoperimetric inequalities forG
~
-
p
,
i
(
K
).Theorem 22.
IfK
∈
S
o
n, p
≥
1, and i
∈
R, then we have the following.
(i) Ifi
≤
0,(99)
G
~
-
p
,
i
K
G
~
-
p
,
i
B
≥
V
K
V
B
n
+
p
n
-
i
/
n
2
,with equality if K is a ball centered at the origin.
(ii) Ifi
≥
n,(100)
G
~
-
p
,
i
K
G
~
-
p
,
i
B
≤
V
K
V
B
n
+
p
n
-
i
/
n
2
,with equality if K is a ball centered at the origin.Proof.
(i) Fori
=
0, by (79) and (24), it follows that (101)
G
~
-
p
K
G
~
-
p
B
≥
V
K
V
B
n
+
p
/
n
.This is Wang’s inequality (24).
Fori
=
n, by (74), (79), and (80), the equality holds trivially in (100).
Fori
<
0, since G
~
-
p
,
i
(
B
)
=
G
~
-
p
(
B
)
=
n
ω
n, by (24), (92), and (94), we have (102)
G
~
-
p
,
i
K
G
~
-
p
,
i
B
≥
G
~
-
p
K
G
~
-
p
B
n
-
i
/
n
.Hence, for i
<
0 and p
≥
1, the L
p-affine isoperimetric inequalities (92) and (24) imply that(103)
G
~
-
p
,
i
K
G
~
-
p
,
i
B
≥
G
~
-
p
K
G
~
-
p
B
n
-
i
/
n
≥
V
K
V
B
n
+
p
n
-
i
/
n
2
,with equality if K is a ball centered at the origin.
(ii) Fori
=
n, by (74), (79), and (80), the equality holds trivially in (100). We now prove the case i
>
n. Inequality (93) and the definition of dual L
p-geominimal surface area give the following:(104)
G
~
-
p
,
i
K
G
~
-
p
,
i
B
≤
G
~
-
p
K
G
~
-
p
B
n
-
i
/
n
≤
V
K
V
B
n
+
p
n
-
i
/
n
2
,with equality if K is a ball centered at the origin.The following results are interesting.Theorem 23.
LetK
∈
S
o
n, p
≥
1, i
∈
R, and 0
≤
i
≤
n; then, (105)
G
~
-
p
,
i
K
G
~
-
p
,
i
B
≤
S
~
p
K
S
~
p
B
n
-
i
/
n
,with equality if K is a ball centered at the origin.Proof.
Inequality (93) and the definition of dual L
p-geominimal surface area give the following:(106)
G
~
-
p
,
i
K
G
~
-
p
,
i
B
n
≤
G
~
-
p
K
n
ω
n
n
-
i
≤
n
V
~
-
p
K
,
B
n
V
~
-
p
B
,
B
n
-
i
=
S
~
p
K
S
~
p
B
n
-
i
,with equality if K is a ball centered at the origin.



---

*Source: 1481793-2016-07-10.xml*