# Orlicz Mean Dual Affine Quermassintegrals

Authors: Chang-Jian Zhao; Wing-Sum Cheung
Journal: Journal of Function Spaces
 (2018)

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## Abstract

Our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first Orlicz variation of the mean dual affine quermassintegrals and call it the Orlicz mean dual affine quermassintegral. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for them are extended to an Orlicz setting. The related concepts and inequalities of dual Orlicz mixed volumes are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield theLp-dual Minkowski inequality and Brunn-Minkowski inequality for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality.


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## Body

## 1. Introduction

The radial additionK+~L of star sets (compact sets that are star-shaped at o and contain o) K and L can be defined by (1)K+~L=x+~y:x∈K,y∈L,where x+~y=x+y if x,y, and o are collinear and x+~y=o, otherwise, or by (2)ρK+~L,·=ρK,·+ρL,·,where ρ(K,·) denotes the radial function of star set K, which is defined by (3)ρK,u=max⁡c≥0:cu∈K,for u∈Sn-1, where Sn-1 is the surface of the unit sphere. Hints as to the origins of the radial addition can be found in [1, p. 235]. If ρ(K,·) is positive and continuous, K will be called a star body. Let Sn denote the set of star bodies about the origin in Rn. When combined with volume, radial addition gives rise to another substantial appendage to the classical theory, called the dual Brunn-Minkowski theory. Radial addition is the basis for the dual Brunn-Minkowski theory (see, e.g., [2–10] for recent important contributions). The original theory is originated from Lutwak [11]. He introduced the concept of dual mixed volume which laid the foundation of the dual Brunn-Minkowski theory. The dual theory can count among its successes the solutions of the Busemann-Petty problem in [3, 4, 9, 12, 13]. For p≠0, x∈Rn, and K,L∈Sn, the p-radial addition K+~pL is defined by (see [14]) (4)ρK+~pL,xp=ρK,xp+ρL,xp.The Lp-harmonic radial combination for star bodies was introduced: If K,L∈Sn, u∈Sn-1, and p≥1, then the Lp-harmonic radial addition defined by Lutwak [8] is(5)ρK+~pL,u-p=ρK,u-p+ρL,u-p.For convex bodies, the Lp-harmonic addition was first investigated by Firey [15].IfK is a nonempty closed (not necessarily bounded) convex set in Rn, then (6)hK,x=max⁡x·y:y∈K,for x∈Rn, which defined the support function h(K,x) of K. A nonempty closed convex set is uniquely determined by its support function. Lp-addition and inequalities are the fundamental and core content in the Lp-Brunn-Minkowski theory. In recent years, a new extension of Lp-Brunn-Minkowski theory is to Orlicz-Brunn-Minkowski theory, initiated by Lutwak et al. [16, 17]. Gardner et al. [18] introduced the Orlicz addition for the first time, constructed a general framework for the Orlicz-Brunn-Minkowski theory, and made the relation to Orlicz spaces and norms clear. The Orlicz addition of convex bodies was also introduced from different angles and the Lp-Brunn-Minkowski inequality was extended to the Orlicz-Brunn-Minkowski inequality (see [19]). The Orlicz centroid inequality for star bodies was introduced in [20]. The other articles advancing the theory can be found in literatures [7, 21–25].Just as theLp-Brunn-Minkowski theory is extended to the Orlicz Brunn-Minkowski theory, it has recently turned to a study extending from Lp-dual Brunn-Minkowski theory to dual Orlicz Brunn-Minkowski theory. The dual Orlicz-Brunn-Minkowski theory has also attracted mathematicians’ attention [14, 26–28]. In 2014, Zhu et al. [29] introduced the Orlicz harmonic radial sum K+~ϕL of two star bodies K and L, defined by (7)ρK+~ϕL,u=sup⁡λ>0:ϕρK,uλ+ϕρL,uλ≤ϕ1,where u∈Sn-1, ϕ:(0,∞)→(0,∞) is a convex and decreasing function such that ϕ(0)=∞, limt→∞ϕ(t)=0, and limt→0ϕ(t)=∞. Let C denote the class of the convex and decreasing functions ϕ. When p≥1 and ϕ(t)=t-p, the Orlicz harmonic addition +~ϕ becomes the Lp-harmonic radial addition +~p. The dual Orlicz mixed volume with respect to Orlicz harmonic radial addition, denoted by V~ϕ(K,L), is defined by (8)V~ϕK,L≔ϕ+′1nlimε→0+VK+~ϕε·L-VKε=1n∫Sn-1ϕρL,uρK,uρK,undSu,where K+~ϕε·L is the Orlicz linear combination of K and L, dS(u) denotes the surface area measure of the unit sphere Sn-1, and ϕ+′(1) denotes the value of the right derivative of convex function ϕ at point 1.The dual affine quermassintegrals were defined, for a convex bodyK∈Sn, by letting Φ~0(K)≔V(K),Φ~n(K)≔ωn, and for 0<j<n (see, e.g., [30], p. 515) (9)Φ~n-jK≔ωn∫Gn,jvoljK∩ξωjndμjξ1/n,where Gn,j denotes the Grassmann manifold of j-dimensional subspaces in Rn, μj denotes the gauge Haar measure on Gn,j, volj(K∩ξ) denotes the j-dimensional volume of intersection of K on j-dimensional subspace ξ⊂Rn, and ωj denotes the volume of j-dimensional unit ball. Gardner [31] showed the Brunn-Minkowski inequality for the dual affine quermassintegrals. If K,L∈Sn and 0≤j≤n-1, then (10)Φ~jK+~L1/n-j≤Φ~jK1/n-j+Φ~jL1/n-j,with equality if and only if K is a dilate of L, modulo a set of measure zero. In analogy to (9), one may also define mean dual affine quermassintegrals by (see, e.g., [30], p. 516) (11)Φ¯n-jK≔ωn∫An,jvoljK∩ξωjn+1dνjξ1/n+1,for a convex body and 0<j<n and by letting Φ¯0(K)≔V(K) and Φ¯n(K)≔ωn. Here, An,j denotes the space of the j-dimensional affine subspace in Rn and νj denotes the gauge Haar measure on An,j. They are related to the dual affine quermassintegrals by (see [32], p. 373). (12)Φ¯n-jK=ωnωj∫KΦ~n-jK-xndx1/n+1.Obviously, Φ¯n-j(K) is invariant under unimodular affine transformations of K.In the paper, our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity such as Orlicz mean dual affine quermassintegrals. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for the mean dual affine quermassintegrals are extended to an Orlicz setting. The new Orlicz-Minkowski and Brunn-Minkowski inequalities for the Orlicz mean dual affine quermassintegrals in special case yield theLp-dual Minkowski inequality and Brunn-Minkowski inequalities for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and Brunn-Minkowski inequalities for general volumes.Following the basic spirit of Alexandroff [33], Fenchel and Jessen [34] introduction of mixed quermassintegrals, and introduction of Lutwak’s Lp-mixed quermassintegrals (see [8, 35]), the study is based on the first-order Orlicz variation of the dual affine quermassintegrals. In Section 3, we prove that the Orlicz first-order variation of the mean dual affine quermassintegrals can be expressed as follows: for ϕ∈C, ε>0, 0<j≤n, and K,L∈Sn, (13)ddεε=0+Φ¯n-jK+~ϕε·L=jϕ+′1Φ¯n-jK-nΦ¯ϕ,n-jK,Ln+1.Putting j=n in (13), then we have the well-known result. (14)ddεε=0+VK+~ϕε·L=nϕ+′1V~ϕK,L.In (13), we find a new geometric quantity. Based on this, we extract the required geometric quantity, denoted by Φ¯ϕ,n-j(K,L), and call it as Orlicz mean dual affine quermassintegrals, defined by (15)Φ¯ϕ,n-jK,L≔ϕ+′1j·Φ¯n-jK-n·ddεε=0+Φ¯n-jK+~ϕε·L1/n+1,where ϕ∈C, 0<j≤n, and K,L∈Sn. We also prove that the new affine geometric quantity Φ¯ϕ,n-j(K,L) has an integral representation. (16)Φ¯ϕ,n-jK,L=ωn∫An,jV~ϕjK∩ξ,L∩ξvoljK∩ξvoljK∩ξωjn+1dνjξ1/n+1,where V~ϕ(j)(K∩ξ,L∩ξ) denotes the Orlicz dual mixed volume of j-dimensional star bodies K∩ξ and L∩ξ in j-dimensional subspace ξ.Obviously, the Orlicz mean dual affine quermassintegrals are an extension of the mean dual affine quermassintegrals; a very natural question is raised: is there a Minkowski type isoperimetric inequality for the Orlicz mean dual affine quermassintegrals? In Section4, we give a positive answer to this question and establish the dual Orlicz-Minkowski inequality for the new affine geometric quantity. For ϕ∈C, 0<j≤n, and K,L∈Sn, we prove the Orlicz-Minkowski inequality for the Orlicz mean dual affine quermassintegrals. (17)Φ¯ϕ,n-jK,LΦ¯n-jKn+1≥ϕΦ¯n-jLΦ¯n-jK1/j.If ϕ is strictly convex, equality holds if and only if K and L are dilates. For j=n, (17) becomes the following dual Orlicz-Minkowski inequality established by Zhu et al. [29]: (18)V~ϕK,L≥VKϕVLVK1/n.If ϕ is strictly convex, equality holds if and only if K and L are dilates.In Section5, on the basis of the dual Minkowski inequality for the Orlicz mean dual affine quermassintegrals, we establish a dual Orlicz-Brunn-Minkowski inequality for the dual mixed mean affine quermassintegrals. If K,L∈Sn, 0<j≤n, and ϕ∈C, then for any ε>0(19)ϕ1≥ϕΦ¯n-jKΦ¯n-jK+~ϕε·L1/j+ε·ϕΦ¯n-jLΦ¯n-jK+~ϕε·L1/j.If ϕ is strictly convex, equality holds if and only if K and L are dilates. For j=n and ε=1, (19) becomes the following dual Orlicz-Brunn-Minkowski inequality established by Zhu et al. [29]. If K,L∈Sn and ϕ∈C, then (20)ϕ1≥ϕVKVK+~ϕL1/n+ϕVLVK+~ϕL1/n.If ϕ is strictly convex, equality holds if and only if K and L are dilates. Moreover, for ε=1, ϕ(t)=t-p, and p≥1, (19) becomes the Lp-dual Brunn-Minkowski inequality for the mean dual affine quermassintegrals. If K,L∈Sn, ε>0, 0<j≤n, p≥1, and ϕ∈C, then (21)Φ¯n-jK+~pL-p/j≥Φ¯n-jK-p/j+Φ¯n-jL-p/j,with equality if and only if K and L are dilates. When j=n, (21) becomes Lutwak’s dual Brunn-Minkowski inequality (36).


## 2. Preliminaries

The setting for this paper isn-dimensional Euclidean space Rn. A body in Rn is a compact set equal to the closure of its interior. For a compact set K⊂Rn, we write V(K) for the (n-dimensional) Lebesgue measure of K and call this the volume of K. Associated with a compact subset K of Rn, which is star-shaped with respect to the origin and contains the origin, its radial function is ρ(K,·):Sn-1→[0,∞), defined by ρ(K,u)=maxλ≥0:λu∈K. Note that the class (star sets) is closed under unions, intersection, and intersection with subspace. The radial function is homogeneous of degree -1; that is, ρ(K,rx)=r-1ρ(K,x), for all x∈Rn and r>0. Let δ~ denote the radial Hausdorff metric, as follows; if K,L∈Sn, then (see, e.g., [30]) (22)δ~K,L=ρK,u-ρL,u∞.From the definition of the radial function, it follows immediately that for g∈GL(n) the radial function of the image gK=gy:y∈K of K is given by (23)ρgK,x=ρK,g-1x,for all x∈Rn.


### 2.1. Dual Mixed Volumes andLp-Dual Mixed Volumes

IfK1,…,Kn∈Sn, the dual mixed volume V~(K1,…,Kn) defined by (see [11]) is as follows: (24)V~K1,…,Kn=1n∫Sn-1ρK1,u⋯ρKn,udSu.If K1=⋯=Kn-i=K,Kn-i+1=⋯=Kn=L, the dual mixed volume V~(K1,…,Kn) is written as V~i(K,L). If K1=⋯=Kn=K, the dual mixed volume V~(K1,…,Kn) is written as V(K). Obviously, For K∈Sn, we have (25)VK=1n∫Sn-1ρK,undSu,and (see [11]) (26)V~1K,L=limε→0+VK+~ε·L-VKε=1n∫Sn-1ρK,un-1ρL,udSu.The fundamental inequality for dual mixed volumes stated that if K,L∈Sn, then (27)V~1K,Ln≤VKn-1VL,with equality if and only if K and L are dilates. The Brunn-Minkowski inequality for the radial addition is the following: If K,L∈Sn, then (28)VK+~L1/n≤VK1/n+VL1/n,with equality if and only if K and L are dilates.The following result follows immediately from the definition ofLp-radial addition, with p≠0. (29)pnlimε→0+VK+~pε·L-VLε=1n∫Sn-1ρK,un-i-pρL,updSu.Let K,L∈Sn and p<0; we define Lp-dual mixed volume of star bodies K and L, V~p(K,L), by (30)V~pK,L=1n∫Sn-1ρK,un-pρL,updSu.This integral representation (30), together with the Hölder inequality, yields the p-dual Minkowski inequality (see [36]): If K,L∈Sn and p<0, then (31)V~pK,Ln≥VKn-pVLp,with equality if and only if K and L are dilates. The definition of Lp-radial addition, together with (31), yields Gardner’s Brunn-Minkowski inequality for p-radial addition (see [37]). If K,L∈Sn and p<0, then (32)VK+~pLp/n≥VKp/n+VLp/n,with equality if and only if K and L are dilates.


### 2.2.Lp-Harmonic Mixed Volumes

The following result follows immediately form (5) with p≥1. (33)-pnlimε→0+VK+~pε·L-VLε=1n∫Sn-1ρK,un+pρL,u-pdSu.Let K,L∈Sn and p≥1; the Lp-harmonic mixed volumes of star bodies K and L denotes V~-p(K,L), defined by (see [35]) (34)V~-pK,L=1n∫Sn-1ρK,un+pρL,u-pdSu.This integral representation (34), together with the Hölder inequality, yields Lutwak’s Lp-dual Minkowski inequality as follows: If K,L∈Sn and p≥1, then (35)V~-pK,Ln≥VKn+pVL-p,with equality if and only if K and L are dilates. This integral representation (34), together with the definition of p-harmonic addition, yields Lutwak’s Lp-Brunn-Minkowski inequality for harmonic p-addition (see [35]). If K,L∈Sn and p≥1, then (36)VK+~pL-p/n≥VK-p/n+VL-p/n,with equality if and only if K and L are dilates.


### 2.3. Orlicz Harmonic Addition and Orlicz Harmonic Linear Combination

Definition 1.
Letm≥2,ϕ∈C, Kj∈Sn, j=1,…,m, define the Orlicz harmonic addition of K1,…,Km, denoted by K1+~ϕ⋯+~ϕKm, defined by (37)ρK1+~ϕ⋯+~ϕKm,x=supλ>0:∑j=1mϕρKj,xλ≤ϕ1,for all x∈Rn.
Equivalently, the Orlicz harmonic additionK1+~ϕ⋯+~ϕKm can be defined implicitly by (38)ϕρK1,xρK1+~ϕ⋯+~ϕKm,x+⋯+ϕρKm,xρK1+~ϕ⋯+~ϕKm,x=ϕ1,for all x∈Rn.
The Orlicz harmonic linear combination on the casem=2 is defined.Definition 2.
Orlicz harmonic linear combination+~ϕ(K,L,α,β) for K,L∈Sn, ϕ∈C, and α,β≥0  (both not zero) is defined by (39)α·ϕρK,xρ+~ϕK,L,α,β,x+β·ϕρL,xρ+~ϕK,L,α,β,x=ϕ1,for all x∈Rn.
Whenϕ(t)=t-p and p≥1, then Orlicz harmonic linear combination +~ϕ(K,L,α,β) changes to the Lp-harmonic linear combination α◊K+~pβ◊L (see [9]). Moreover, we shall write K+~ϕε·L instead of +~ϕ(K,L,1,ε), for ε≥0, and assume throughout that this is defined by (39), where α=1,β=ε, and ϕ∈C, and write +~ϕ(K,L,1,1) as K+~ϕL.


## 2.1. Dual Mixed Volumes andLp-Dual Mixed Volumes

IfK1,…,Kn∈Sn, the dual mixed volume V~(K1,…,Kn) defined by (see [11]) is as follows: (24)V~K1,…,Kn=1n∫Sn-1ρK1,u⋯ρKn,udSu.If K1=⋯=Kn-i=K,Kn-i+1=⋯=Kn=L, the dual mixed volume V~(K1,…,Kn) is written as V~i(K,L). If K1=⋯=Kn=K, the dual mixed volume V~(K1,…,Kn) is written as V(K). Obviously, For K∈Sn, we have (25)VK=1n∫Sn-1ρK,undSu,and (see [11]) (26)V~1K,L=limε→0+VK+~ε·L-VKε=1n∫Sn-1ρK,un-1ρL,udSu.The fundamental inequality for dual mixed volumes stated that if K,L∈Sn, then (27)V~1K,Ln≤VKn-1VL,with equality if and only if K and L are dilates. The Brunn-Minkowski inequality for the radial addition is the following: If K,L∈Sn, then (28)VK+~L1/n≤VK1/n+VL1/n,with equality if and only if K and L are dilates.The following result follows immediately from the definition ofLp-radial addition, with p≠0. (29)pnlimε→0+VK+~pε·L-VLε=1n∫Sn-1ρK,un-i-pρL,updSu.Let K,L∈Sn and p<0; we define Lp-dual mixed volume of star bodies K and L, V~p(K,L), by (30)V~pK,L=1n∫Sn-1ρK,un-pρL,updSu.This integral representation (30), together with the Hölder inequality, yields the p-dual Minkowski inequality (see [36]): If K,L∈Sn and p<0, then (31)V~pK,Ln≥VKn-pVLp,with equality if and only if K and L are dilates. The definition of Lp-radial addition, together with (31), yields Gardner’s Brunn-Minkowski inequality for p-radial addition (see [37]). If K,L∈Sn and p<0, then (32)VK+~pLp/n≥VKp/n+VLp/n,with equality if and only if K and L are dilates.


## 2.2.Lp-Harmonic Mixed Volumes

The following result follows immediately form (5) with p≥1. (33)-pnlimε→0+VK+~pε·L-VLε=1n∫Sn-1ρK,un+pρL,u-pdSu.Let K,L∈Sn and p≥1; the Lp-harmonic mixed volumes of star bodies K and L denotes V~-p(K,L), defined by (see [35]) (34)V~-pK,L=1n∫Sn-1ρK,un+pρL,u-pdSu.This integral representation (34), together with the Hölder inequality, yields Lutwak’s Lp-dual Minkowski inequality as follows: If K,L∈Sn and p≥1, then (35)V~-pK,Ln≥VKn+pVL-p,with equality if and only if K and L are dilates. This integral representation (34), together with the definition of p-harmonic addition, yields Lutwak’s Lp-Brunn-Minkowski inequality for harmonic p-addition (see [35]). If K,L∈Sn and p≥1, then (36)VK+~pL-p/n≥VK-p/n+VL-p/n,with equality if and only if K and L are dilates.


## 2.3. Orlicz Harmonic Addition and Orlicz Harmonic Linear Combination

Definition 1.
Letm≥2,ϕ∈C, Kj∈Sn, j=1,…,m, define the Orlicz harmonic addition of K1,…,Km, denoted by K1+~ϕ⋯+~ϕKm, defined by (37)ρK1+~ϕ⋯+~ϕKm,x=supλ>0:∑j=1mϕρKj,xλ≤ϕ1,for all x∈Rn.
Equivalently, the Orlicz harmonic additionK1+~ϕ⋯+~ϕKm can be defined implicitly by (38)ϕρK1,xρK1+~ϕ⋯+~ϕKm,x+⋯+ϕρKm,xρK1+~ϕ⋯+~ϕKm,x=ϕ1,for all x∈Rn.
The Orlicz harmonic linear combination on the casem=2 is defined.Definition 2.
Orlicz harmonic linear combination+~ϕ(K,L,α,β) for K,L∈Sn, ϕ∈C, and α,β≥0  (both not zero) is defined by (39)α·ϕρK,xρ+~ϕK,L,α,β,x+β·ϕρL,xρ+~ϕK,L,α,β,x=ϕ1,for all x∈Rn.
Whenϕ(t)=t-p and p≥1, then Orlicz harmonic linear combination +~ϕ(K,L,α,β) changes to the Lp-harmonic linear combination α◊K+~pβ◊L (see [9]). Moreover, we shall write K+~ϕε·L instead of +~ϕ(K,L,1,ε), for ε≥0, and assume throughout that this is defined by (39), where α=1,β=ε, and ϕ∈C, and write +~ϕ(K,L,1,1) as K+~ϕL.


## 3. Orlicz Mean Dual Affine Quermassintegrals

In order to define Orlicz mean dual affine quermassintegrals, we need the following lemmas.Lemma 3.
IfK,L∈Sn and ϕ∈C, then for ε>0(40)ϕ1VK+~ϕε·L=V~ϕK+~ϕε·L,K+ε·V~ϕK+~ϕε·L,L.Proof.
From (8) and (39), we have for any Q∈Sn(41)V~ϕQ,K+ε·V~ϕQ,L=1n∫Sn-1φρK,uρQ,u+ε·ϕρL,uρQ,uρQ,undSu=ϕ1n∫Sn-1ρQ,undSu=ϕ1VQ.
PuttingQ=K+~ϕε·L in (41), (41) easily becomes (40).Lemma 4 (see [29]).
IfK,L∈Sn and ϕ∈C, then for ε>0(42)K+~ϕε·L⟶K,in the radial Hausdorff metric as ε→0+.Lemma 5.
IfK,L∈Sn, ε>0, and ϕ∈C, then (43)K+~ϕε·L∩ξ=K∩ξ+~ϕε·L∩ξ.Proof.
Supposeξ∈An,j and Sj-1=Sn-1∩ξ. For u∈Sj-1 and Q∈Sn, we have (44)ρQ,u=ρQ∩ξ,u.Hence (45)ϕρK∩ξ,uρK+~ϕε·L∩ξ,u+εϕρL∩ξ,uρK+~ϕε·L∩ξ,u=ϕ1.On the other hand (46)ϕρK∩ξ,uρK∩ξ+~ϕε·L∩ξ,u+εϕρL∩ξ,uρK∩ξ+~ϕε·L∩ξ,u=ϕ1.Therefore (K+~ϕε·L)∩ξ and (K∩ξ)+~ϕε·(L∩ξ) are the same star body in ξ.Definition 6.
Ifϕ∈C, 0<j≤n, and K,L∈Sn, then Orlicz mean dual affine quermassintegral of K and L, denoted by Φ¯ϕ,n-j(K,L), is defined by (47)Φ¯ϕ,n-jK,L≔ωn∫An,jV~ϕjK∩ξ,L∩ξvoljK∩ξvoljK∩ξωjn+1dνjξ1/n+1.Specifically, we agreed on the following: (48)Φ¯ϕ,0K,L=V~ϕK,LVK1/n+1VK.
In order to define the Orlicz mean dual affine quermassintegrals, we need also to calculate the first Orlicz variation of the mean dual affine quermassintegrals.Lemma 7.
Ifϕ∈C, 0<j≤n, and K,L∈Sn, then for any ε>0(49)ddεε=0+Φ¯n-jK+~ϕε·L=jϕ+′1Φ¯n-jK1-nΦ¯ϕ,n-jK,Ln.Proof.
On the one hand, from (8), we have (50)ddεε=0+∫An,jvoljK+~ϕε·L∩ξn+1dνjξ=limε→0+∫An,jvoljK+~ϕε·L∩ξn+1-voljK∩ξn+1εdνjξ=n+1limε→0+∫An,jvoljK∩ξn·ddεε=0+voljK+~ϕε·L∩ξdνjξ=n+1jϕ+′1∫An,jvoljK∩ξnV~ϕjK∩ξ,L∩ξdνjξ.On the other hand, from (11), (47), and (50), we obtain (51)ddεε=0+Φ¯n-jK+~ϕε·L=ωnωj·ddεε=0+∫An,jvoljK+~ϕε·L∩ξn+1dνjξ1/n+1=ωnn+1ωj∫An,jvoljK∩ξn+1dνjξ-n/n+1ddεε=0+∫An,jvoljK+~ϕε·L∩ξn+1dνjξ=jϕ+′1ωnωj∫An,jvoljK∩ξndνjξ-n/n+1∫An,jV~ϕjK∩ξ,L∩ξvoljK∩ξvoljK∩ξn+1dνjξ=jϕ+′1Φ¯n-jK-nΦ¯ϕ,n-jK,Ln+1.Lemma 8.
IfK,L∈Sn, 0<j≤n, and ϕ∈C, then (52)Φ¯ϕ,n-jK,K=ϕ11/n+1Φ¯n-jK.Proof.
The definition of the Orlicz mean dual affine quermassintegrals, together with (8) and (11), gives (52).
Ifϕ(t)=t-p,p≥1, then Φ¯ϕ,n-j(K,L)=Φ¯-p,n-j(K,L) and call the Lp-dual mixed mean affine quermassintegral of K and L, and (53)Φ¯-p,n-jK,L=ωn∫An,jV~-pjK∩ξ,L∩ξvoljK∩ξvoljK∩ξωjn+1dνjξ1/n+1,where V~-p(j)(K∩ξ,L∩ξ) denotes the Lp-dual mixed volume of j-dimensional star bodies K∩ξ and L∩ξ in j-dimensional subspace ξ.Lemma 9 (see [29]).
IfK,L∈Sn, ϕ∈C, and any g∈GL(n), then for ε>0(54)gK+~ϕε·L=gK+~ϕε·gL.In the following, we will prove that Orlicz mean dual affine quermassintegralΦ¯ϕ,n-j(K,L) is invariant under simultaneous unimodular centro-affine transformation.Lemma 10.
IfK,L∈Sn, 0<j≤n,ϕ∈C, and any g∈SL(n), then (55)Φ¯ϕ,n-jgK,gL=Φ¯ϕ,n-jK,L.Proof.
Suppose thatξ∈An,j and Sj-1=Sn-1∩ξ. For any g∈SL(n), u∈Sj-1, and Q∈Sn, we have (56)ρgQ,u=ρgQ∩ξ,u.When x∈Rn∖0, let (57)x=xx.From (23), we obtain (58)V~ϕjgK∩ξ,gL∩ξ=1j∫Sn-1∩ξϕρgL∩ξ,uρgK∩ξ,uρgK∩ξ,ujdSu=1j∫Sn-1ϕρL,g-1uρK,g-1uρK,g-1ujdSg-1u=1j∫Sn-1∩ξϕρL∩ξ,g-1uρK∩ξ,g-1uρK∩ξ,g-1ujdSg-1u=V~ϕjK∩ξ,L∩ξ.On the other hand, from Definition 6 and (58), we have (59)Φ¯ϕ,n-jgK,gL=ωn∫An,jV~ϕjgK∩ξ,gL∩ξvoljgK∩ξvoljgK∩ξωjn+1dνjξ1/n+1=ωn∫An,jV~ϕjK∩ξ,L∩ξvoljK∩ξvoljK∩ξωjn+1dνjξ1/n+1=Φ¯ϕ,n-jK,L.Next, we also can give another proof directly.Proof.
From Lemmas7 and 9, we have, for g∈SL(n), (60)Φ¯ϕ,n-jgK,gL=ϕ+′1jΦ¯n-jgK-n·ddεε=0+Φ¯n-jgK+~ϕε·gL1/n+1=ϕ+′1jΦ¯n-jgK-n·ddεε=0+Φ¯n-jgK+~ϕε·L1/n+1=ϕ+′1jΦ¯n-jK-n·ddεε=0+Φ¯n-jK+~ϕε·L1/n+1=Φ¯ϕ,n-jK,L.Here, we point out the connections between the Orlicz mean dual affine quermassintegrals and the dual affine quermassintegrals. From (13) and in view of the connections between the mean dual affine quermassintegrals and the dual affine quermassintegrals, we have the following: for ϕ∈C, ε>0, 0<j≤n, and K,L∈Sn, (61)Φ¯ϕ,n-jK,Ln+1=ϕ+′1ωnnjωjn·ddεε=0+Φ¯n-jK+~ϕε·L·∫KΦ~n-jK-xndxn/n+1.We also need the following lemma to prove our main results.Lemma 11 (Jensen’s inequality).
Letμ be a probability measure on a space X and g:X→I⊂R is a μ-integrable function, where I is a possibly infinite interval. If ψ:I→R is a convex function, then (62)∫Xψgxdμx≥ψ∫Xgxdμx.If ψ is strictly convex, equality holds if and only if g(x) is constant for μ-almost all x∈X (see [38, p.165]).


## 4. Orlicz-Minkowski Inequality for Orlicz Mean Dual Quermassintegrals

Theorem 12.
IfK,L∈Sn, ϕ∈C, and 0<j≤n, then (63)Φ¯ϕ,n-jK,LΦ¯n-jKn+1≥ϕΦ¯n-jLΦ¯n-jK1/j.If ϕ is strictly convex, equality holds if and only if K and L are dilates.Proof.
Whenj=n, (63) becomes the dual Orlicz-Minkowski inequality; hence we assume 0<j<n. Since (64)∫An,jdνξ=∫An,jvoljK∩ξn+1∫An,jvoljK∩ξn+1dνjξdνjξ=1,the above equation defines a Borel probability measure ν on An,j; namely, (65)dνξ=voljK∩ξn+1∫An,jvoljK∩ξn+1dνjξdνjξ.
From (11), (47), and (65) and using dual Orlicz-Minkowski inequality, Jensen inequality, and Hölder inequality, we obtain (66)Φ¯ϕ,n-jK,LΦ¯n-jKn+1=∫An,jV~ϕjK∩ξ,L∩ξ/voljK∩ξvoljK∩ξ/ωjn+1dνjξ∫An,jvoljK∩ξ/ωjn+1dνjξ=∫An,jV~ϕjK∩ξ,L∩ξvoljK∩ξdν≥∫An,jϕvoljL∩ξvoljK∩ξ1/jdν≥ϕ∫An,jvoljL∩ξvoljK∩ξ1/jdν=ϕ∫An,jvoljK∩ξjn+1-1/jvoljL∩ξ1/jdνjξ∫An,jvoljK∩ξn+1dνjξ≥ϕ∫An,jvoljK∩ξn+1dνjξjn+1-1/jn+1∫An,jvoljL∩ξn+1dνjξ1/jn+1∫An,jvoljK∩ξn+1dνjξ=ϕΦ¯n-jLΦ¯n-jK1/j.Next, we discuss the equal condition of (63). If ϕ is strictly convex, suppose that K and L are dilates; that is, there exists λ>0 such that L=λK. Hence (67)Φ¯ϕ,n-jK,LΦ¯n-jKn+1=Φ¯ϕ,n-jK,λKΦ¯n-jKn+1=ϕλ1/n+1Φ¯n-jKΦ¯n-jKn+1=ϕλ=ϕΦ¯n-jλKΦ¯n-jK1/j=ϕΦ¯n-jLΦ¯n-jK1/j.This implies that the equality in (63) holds.
On the other hand, suppose the equality holds in (63); then these three inequalities in the above proof must satisfy the equal sign. Since the first inequality in the above proof is the dual Orlicz-Minkowski inequality, (68)V~ϕjK∩ξ,L∩ξvoljK∩ξ≥ϕvoljL∩ξvoljK∩ξ1/j.Form the equality condition of dual Orlicz-Minkowski inequality, if the equality holds, then K∩ξ and L∩ξ must be dilates. The second inequality in the above proof is Jensen inequality. (69)∫An,jϕvoljL∩ξvoljK∩ξ1/jdν≥ϕ∫An,jvoljL∩ξvoljK∩ξ1/jdν.From the equality condition of Jensen inequality, if ϕ is strictly convex and the equality holds, then volj(L∩ξ)/volj(K∩ξ) must be a constant; this yields that K∩ξ and L∩ξ must be dilates. In this proof, the third inequality is obtained by applying the Hölder inequality. From the equality condition of Hölder inequality, this yields that equality holds and volj(K∩ξ) and volj(L∩ξ) must be proportional; namely, K∩ξ and L∩ξ are dilates. From the combinations of these equal conditions, it follows that equality in (63) holds, if ϕ is strictly convex, and equality holds if and only if K and L are dilates.Corollary 13.
IfK,L∈Sn, p≥1, and 0<j≤n, then (70)Φ¯-p,n-jK,LΦ¯n-jKn+1≥Φ¯n-jLΦ¯n-jK-p/j,with equality if and only if K and L are dilates.Proof.
This follows immediately from (63) with ϕ(t)=t-p and 1<p<∞.Corollary 14.
IfK,L∈Sn and ϕ∈C, then (71)V~ϕK,L≥VKϕVLVK1/n.If ϕ is strictly convex, equality holds if and only if K and L are dilates.Proof.
This follows immediately from (63) with j=n.The following uniqueness is a direct consequence of the Orlicz-Minkowski inequality for the Orlicz mean dual affine quermassintegrals.Theorem 15.
Ifϕ∈C and is strictly convex, 0<j≤n and M⊂Sn such that K,L∈M. If (72)Φ¯ϕ,n-jM,K=Φ¯ϕ,n-jM,L,∀M∈Mor (73)Φ¯ϕ,n-jK,MΦ¯n-jK=Φ¯ϕ,n-jL,MΦ¯n-jL,∀M∈Mthen K=L.Proof.
Suppose that (72) holds. Taking K for M, then, from Lemma 8 and (63), we obtain (74)ϕ1Φ¯n-jKn+1=Φ¯ϕ,n-jK,Ln+1≥Φ¯n-jKn+1ϕΦ¯n-jLΦ¯n-jK1/j,with equality if and only if K and L are dilates. Hence (75)ϕ1≥ϕΦ¯n-jLΦ¯n-jK1/j,with equality if and only if K and L are dilates. Since ϕ is decreasing function on (0,∞), it follows that (76)Φ¯n-jK≤Φ¯n-jL,with equality if and only if K and L are dilates. On the other hand, if taking L for M, we similarly get Φ¯n-j(K)≥Φ¯n-j(L), with equality if and only if K and L are dilates. Hence Φ¯n-j(K)=Φ¯n-j(L), and K and L are dilates; it follows that K and L must be equal.
Suppose that (73) holds. Taking L for M, then, from Lemma 8 and (63), we obtain (77)ϕ1=Φ¯ϕ,n-jK,Ln+1Φ¯n-jKn+1≥ϕΦ¯n-jLΦ¯n-jK1/j,with equality if and only if K and L are dilates. Hence (78)ϕ1≥ϕΦ¯n-jLΦ¯n-jK1/j,with equality if and only if K and L are dilates. Since ϕ is decreasing function on (0,∞), it follows that (79)Φ¯n-jK≤Φ¯n-jL,with equality if and only if K and L are dilates. On the other hand, if we take K for M, we similarly get Φ¯n-j(K)≥Φ¯n-j(L), with equality if and only if K and L are dilates. Hence Φ¯n-j(K)=Φ¯n-j(L), and K and L are dilates; it follows that K and L must be equal.Corollary 16 (see [29]).
Ifϕ∈C and is strictly convex, 0<j≤n and M⊂Sn such that K,L∈M. If (80)V~ϕM,K=V~ϕM,L,∀M∈Mor (81)V~ϕK,MVK=V~ϕL,MVL,∀M∈Mthen K=L.Proof.
This follows immediately from Theorem15 with j=n.


## 5. Orlicz-Brunn-Minkowski Inequality for Mean Dual Affine Quermassintegrals

Lemma 17.
IfK,L∈Sn, 0<j≤n, and ϕ∈C, then for any ε>0(82)ϕ1=Φ¯ϕ,n-jK+~ϕε·L,KΦ¯n-jK+~ϕε·Ln+1+ε·Φ¯ϕ,n-jK+~ϕε·L,LΦ¯n-jK+~ϕε·Ln+1.Proof.
From (8) and Lemmas 3 and 5, we have (83)V~ϕK+~ϕε·L∩ξ,K∩ξ+εV~ϕK+~ϕε·L∩ξ,L∩ξ=V~ϕK∩ξ+~ϕε·L∩ξ,K∩ξ+εV~ϕK∩ξ+~ϕε·L∩ξ,L∩ξ=V~ϕK∩ξ+~ϕε·L∩ξ,K∩ξ+~ϕε·L∩ξ=ϕ1voljK∩ξ+~ϕε·L∩ξ=ϕ1voljK+~ϕε·L∩ξ.Let Q=K+~ϕε·L; from (83) and (47), we have (84)Φ¯ϕ,n-jQ,Kn+1+ε·Φ¯ϕ,n-jQ,Ln+1=ωnn+1∫An,jV~ϕjQ∩ξ,K∩ξ+ε·V~ϕjQ∩ξ,L∩ξvoljQ∩ξvoljQ∩ξωjn+1dνjξ=ϕ1ωnn+1∫An,jvoljQ∩ξωjn+1dνjξ=ϕ1Φ¯n-jQn+1.The proof is complete.Lemma 18.
LetK,L∈Sn, ε>0, and ϕ∈C.(1)
IfK and L are dilates, then K and K+~ϕε·L are dilates.(2)
IfK and K+~ϕε·L are dilates, then K and L are dilates.Proof.
Suppose that there exists a constantλ>0 such that L=λK; we have (85)ϕρK,uρK+~ϕε·L,u+εϕλρK,uρK+~ϕε·L,u=ϕ1.On the other hand, there exists a unique constant δ>0 such that (86)ϕρK,uρδK,u+εϕλρK,uρδK,u=ϕ1,where δ satisfies that (87)ϕ1δ+εϕεδ=ϕ1.This shows that K+~ϕε·L=δK.
Suppose that there exists a constantλ>0 such that K+~ϕε·L=λK. Then (88)ϕ1λ+εϕρL,uρK+~ϕε·L,u=ϕ1.This shows that (89)ρL,uρK+~ϕε·L,uis a constant. This yields that K+~ϕε·L and L are dilates. Namely, K and L are dilates.Theorem 19.
IfK,L∈Sn, ε>0, 0<j≤n, and ϕ∈C, then (90)ϕ1≥ϕΦ¯n-jKΦ¯n-jK+~ϕε·L1/j+ε·ϕΦ¯n-jLΦ¯n-jK+~ϕε·L1/j.If ϕ is strictly convex, equality holds if and only if K and L are dilates.Proof.
From Lemma17 and (63), we obtain (91)ϕ1=Φ¯ϕ,n-jK+~ϕε·L,KΦ¯n-jK+~ϕε·Ln+1+ε·Φ¯ϕ,n-jK+~ϕε·L,LΦ¯n-jK+~ϕε·Ln+1≥ϕΦ¯n-jKΦ¯n-jK+~ϕε·L1/j+ε·ϕΦ¯n-jLΦ¯n-jK+~ϕε·L1/j.If ϕ is strictly convex, from equality condition of the Orlicz-Minkowski inequality, the equality holds if and only if K and K+~ϕε·L are dilates, and L and K+~ϕε·L are dilates and, combining with Lemma 18, this yields that if ϕ is strictly convex, equality holds in (90) if and only if K and L are dilates.Corollary 20.
IfK,L∈Sn, p≥1, ε>0, and 0<j≤n, then (92)Φ¯n-jK+~pε·L-p/j≥Φ¯n-jK-p/j+ε·Φ¯n-jL-p/j,with equality if and only if K and L are dilates.Proof.
This follows immediately from (90) with ϕ(t)=t-p and p≥1.Forj=n and ε=1, (92) becomes Lutwak’s Lp dual Brunn-Minkowski inequality (36).Corollary 21.
IfK,L∈Sn and ϕ∈C, then (93)1≥ϕVKVK+~ϕL1/n+VLVK+~ϕL1/n.If ϕ is strictly convex, equality holds if and only if K and L are dilates.Proof.
This follows immediately from (90) with ε=1 and j=n.Theorem 22.
Orlicz-Minkowski inequality for the Orlicz mean dual affine quermassintegrals is equivalent to the Orlicz Brunn-Minkowski inequality for the mean dual affine quermassintegrals. Namely, ifϕ∈C, 0<j≤n, and K,L∈Sn, then (94)Φ¯ϕ,n-jK,LΦ¯n-jKn+1≥ϕΦ¯n-jLΦ¯n-jK1/j⟺ϕ1≥ϕΦ¯n-jKΦ¯n-jK+~ϕL1/j+ϕΦ¯n-jLΦ¯n-jK+~ϕL1/j.If ϕ is strictly convex, equality holds if and only if K and L are dilates.Proof.
⇐: Let(95)Kε=K+~ϕε·L.
From Lemmas4 and 7 and using the Orlicz-Brunn-Minkowski inequality (90), we obtain (96)jϕ-′1Φ¯n-jK-nΦ¯ϕ,n-jK,Ln+1=ddεε=0+Φ¯n-jKε=limε→0+Φ¯n-jKε-Φ¯n-jKε=limε→0+1-Φ¯n-jK/Φ¯n-jKεϕ1-ϕΦ¯n-jK/Φ¯n-jKε1/j·ϕ1-ϕΦ¯n-jK/Φ¯n-jKε1/jε·Φ¯n-jKε=limt→1+1-tϕ1-ϕt1/j·limε→0+ϕ1-ϕΦ¯n-jK/Φ¯n-jKε1/jε·limε→0+Φ¯n-jKε≥jϕ+′1·limε→0+ϕΦ¯n-jLΦ¯n-jKε1/j·limε→0+Φ¯n-jKε=jϕ+′1·ϕΦ¯n-jLΦ¯n-jK1/j·Φ¯n-jK.
⇒: From the proof of Theorem 19, we may see that Orlicz-Minkowski inequality for Orlicz mean dual affine quermassintegrals implies also Orlicz-Brunn-Minkowski inequality for the mean dual affine quermassintegrals.
This proof is complete.Corollary 23.
Ifϕ∈C and K,L∈Sn, then (97)V~ϕK,LVK≥ϕVLVK1/n⟺ϕ1≥ϕVKVK+~ϕL1/n+ϕVLVK+~ϕL1/n.If ϕ is strictly convex, equality holds if and only if K and L are dilates.Proof.
This follows immediately from Theorem22 with ε=1 and j=n.Corollary 24.
If0<j≤n, p≥1, and K,L∈Sn, then (98)Φ¯-p,n-jK,LΦ¯n-jKn+1≥Φ¯n-jLΦ¯n-jK-p/j⟺Φ¯n-jK+~ϕL-p/j≥Φ¯n-jK-p/j+Φ¯n-jL-p/j.If ϕ is strictly convex, equality holds if and only if K and L are dilates.Proof.
This follows immediately from Theorem22 with ϕ(t)=t-p and p≥1.



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*Source: 8123924-2018-02-01.xml*