# Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Authors: Chang-Jian Zhao
Journal: Journal of Function Spaces
 (2018)

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## Abstract

Our main aim is to generalize the classical mixed volumeV(K1,…,Kn) and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call itOrlicz multiple mixed volume of convex bodies K1,…,Kn, and Ln, denoted by Vφ(K1,…,Kn,Ln), which involves (n+1) convex bodies in Rn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities of Lp-multiple mixed volume Vp(K1,…,Kn,Ln) are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yields Lp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.


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## Body

## 1. Introduction

One of the most important operations in geometry is vector addition. As an operation between setsK and L, defined by (1)K+L=x+y:x∈K,y∈L,it is usually called Minkowski addition, and combined with volume, plays an important role in the Brunn-Minkowski theory. During the last few decades, the theory has been extended to Lp-Brunn-Minkowski theory. The set, called Lp addition, was introduced by Firey in [1, 2]. The operation, denoted by +p, is defined by (2)hK+pL,xp=hK,xp+hL,xp,for all x∈Rn, 1≤p≤∞, and compact convex sets K and L in Rn containing the origin. When p=∞, (2) is interpreted as h(K+∞L,x)=max⁡{h(K,x),h(L,x)}, as is customary. Here the functions are the support functions. If K is a nonempty closed (not necessarily bounded) convex set in Rn, then (3)hK,x=max⁡x·y:y∈K,for x∈Rn, defines the support function h(K,x) of K. Lp addition, volume, and combine inequalities lead to the Lp-Brunn-Minkowski theory, which are the fundamental and core content in the Lp Brunn-Minkowski theory. For some important results and more information from this theory, we refer to [3–22] and the references therein.L p spaces have a natural generalization, known as the Orlicz spaces. In recent years, progress towards an Orlicz-Brunn-Minkowski theory is initiated by Lutwak, Yang, and Zhang [23, 24] to initiate an extension of the Lp Brunn-Minkowski theory to an Orlicz-Brunn-Minkowski theory. They successively established the fundamental affine inequalities for these bodies. The definition of a corresponding addition is shown in later work of Gardner, Hug, and Weil [25] and they developed a very general and comprehensive Orlicz-Brunn-Minkowski theory. And they constructed a general framework for the Orlicz-Brunn-Minkowski theory and made clear for the first time the relation to Orlicz spaces and norms. The Orlicz addition of convex bodies was also introduced from different angles and established the Orlicz-Brunn-Minkowski inequality for the Orlicz addition (see [26]). The Orlicz centroid inequality for star bodies was introduced which is an extension from convex to star bodies (see [27]). The existence of even Orlicz-Minkowski problem is demonstrated by Haberl, Lutwak, Yang, and Zhang [28]. Ludwig [29], Ludwig, and Reitzner [11] introduced what soon came to be seen as the Orlicz affine area; for related work, see [30]. In these papers the notions of Lp-addition, Lp-mixed volume, Lp-affine surface area, Lp-centroid body, and Lp-projection body and inequalities were extended to an Orlicz setting. Advances in the theory and its dual can be found in [31–48].In 2014, Gardner, Hug, and Weil [25] introduced the Orlicz addition K+φL of compact convex sets K and L in Rn containing the origin, implicitly, by (4)hK+φL,u=inf⁡λ>0:φhK,uλ+φhL,uλ≤1,where φ:[0,∞)→(0,∞) is a convex and increasing function such that φ(0)=0 and φ(1)=1. Let Φ denote the set of convex functions φ:[0,∞)→[0,∞) that is increasing and satisfies φ(0)=0 and φ(1)=1. For φ(t)=tp and p≥1, the new addition K+φL becomes K+pL. The Orlicz mixed volume with respect to the Orlicz addition, denoted by Vφ(K,L), is defined by (5)VφK,L=1n∫Sn-1φhL,uhK,uhK,udSK;u,where S(K;u) is the surface area measure of K, K is a convex body containing the origin in its interior, and L is a compact convex set containing the origin. For φ(t)=tp and p≥1, Vφ(K,L) becomes Vp(K,L), defined by (see [49]) (6)VpK,L=1n∫Sn-1hL,uphK,u1-pdSK;u.Associated with the convex bodies K1,…,Kn-1 in Rn is a unique positive Borel measure on Sn-1, S(K1,…,Kn-1;·); call it the mixed area measure of K1,…,Kn-1. For any convex body Kn, one has the integral representation (see, e.g., [5], p. 354). (7)VK1,…,Kn=1n∫Sn-1hKn,udSK1,…,Kn-1;u.The integration is with respect to the mixed area measure S(K1,…,Kn-1;·) on Sn-1. The mixed area measure S(K1,…,Kn-1;·) is symmetric in its (first n-1) arguments. When K1=⋯=Kn-i-1=K and Kn-i=⋯=Kn-1=B, the mixed area measure S(K,…,K,B,…,B;·) with i copies of B and (n-i-1) copies of K will be written as Si(K,·). For K1=⋯=Kn-1=K, S(K1,…,Kn-1;·) reduces to the surface area measure S(K;·). The classical Aleksandrov-Fenchel inequality is as follows: if K1,…,Kn are convex bodies containing the origin and 1≤r≤n (see, e.g., [50, p. 401]), then (8)VK1,…,Kn≥∏i=1rVKi…,Ki,Kr+1,…,Kn1/r.In the paper, we further improve the Orlicz-Brunn-Minkowski theory. Our main aim is to introduce a new mixed volume of convex bodiesK1,…,Kn,Ln, denoted as Vφ(K1,…,Kn,Ln), which involves (n+1) convex bodies in Rn, and call it Orlicz multiple mixed volume for convex bodies K1,…,Kn, and Ln. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The new Orlicz-Aleksandrov-Fenchel inequality in special cases yields Lp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequality. As application, we prove Orlicz-Brunn-Minkowski inequality for the mixed volumes, which implies the Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.Following the spirit of introduction of Aleksandrov [51], Fenchel and Jessen’s mixed quermassintegrals (see [52]), and introduction of Lutwak’s Lp-mixed quermassintegrals (see [49]), we are based on the study of Orlicz first-order variational of mixed volumes. In Section 4, we prove that the Orlicz first-order variation of mixed volumes can be expressed as follows: for φ∈Φ and ε>0(9)ddεε=0+VK1,…,Kn-1,Ln+φ,εKn=1φl′1·VφK1,…,Kn,Ln,where K1,…,Kn are convex bodies containing the origin, Ln is a convex body containing the origin in its interior, V(K1,…,Kn) is the usual mixed volume, and φl′(1) denotes the value of left derivative of convex function φ at point 1. Here, K+φ,εL denotes the Orlicz linear combination of convex bodies K and L. If K,L are convex bodies containing the origin, α,β≥0, and φ∈Φ, then Orlicz linear combination +φ(K,L,α,β) is defined by ([25]) (10)α·φhK,xh+φK,L,α,β,x+β·φhL,xh+φK,L,α,β,x=1.For α=1 and β=ε≥0, the Orlicz linear combination +φ(K,L,1,ε) is denoted by K+φ,εL. This first-order variational equation then provides the new geometric quantity Vφ(K1,…,Kn,Ln); it is called it Orlicz multiple mixed volume of convex bodies K1,…,Kn,Ln and defined by (11)VφK1,…,Kn,Ln≔φl′1·ddεε=0+VK1,…,Kn-1,Ln+φ,εKn.We prove also the new affine geometric quantity Vφ(K1,…,Kn,Ln) which has an integral representation.(12)VφK1,…,Kn,Ln=1n∫Sn-1φhKn,uhLn,uhLn,udSK1,…,Kn-1;u.When φ(t)=tp and p≥1, Vφ(K1,…,Kn,Ln) becomes a new mixed volume Vp(K1,…,Kn,Ln); call it Lp-multiple mixed volume, which is first published and named here. (13)VpK1,…,Kn,Ln=1n∫Sn-1hKn,uphLn,u1-pdSK1,…,Kn-1;u.For p=1, Vp(K1,…,Kn,Ln) becomes V(K1,…,Kn), and for K1=⋯=Kn-i-1=K, Kn-i=⋯=Kn-1=B, Kn=L and Ln=K, Vp(K1,…,Kn,Ln) becomes the p-mixed quermassintegral Wp,i(K,L) of K and L, defined by, for all 0≤i<n (see [49]), (14)Wp,iK,L=1n∫Sn-1hL,uphK,u1-pdSiK;u.Putting Kn=L, Ln=K, and K1=⋯=Kn-1=K in (12), Vφ(K1,…,Kn,Ln) becomes the well-know Orlicz mixed volume Vφ(K,L). This shows that the well-known mixed volumes Vp(K,L), Wp,i(K,L), and Vφ(K,L) are all the special cases of Vφ(K1,…,Kn,Ln), where p≥1 and φ∈Φ. In fact, the Orlicz mixed quermassintegral Wφ,i(K,L) is also a special case of Vφ(K1,…,Kn,Ln). The Orlicz mixed quermassintegral of K and L, Wφ,i(K,L), is defined by (see Section 4) (15)Wφ,iK,L=1n∫Sn-1φhL,uhK,uhK,udSiK,u.In Section5, we establish the following Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed volume. If K1,…,Kn are convex bodies containing the origin, Ln is a convex body containing the origin in its interior, 1≤r≤n, and φ∈Φ, then (16)VφK1,…,Kn,Ln≥VK1,…,Kn-1,Ln·φ∏i=1rVKi…,Ki,Kr+1,…,Kn1/rVK1,…,Kn-1,Ln.When φ(t)=tp and p≥1, (16) becomes a new Lp-Aleksandrov-Fenchel type inequality. If K1,…,Kn are convex bodies containing the origin, Ln is a convex body containing the origin in its interior, 1≤r≤n, and p≥1, then (17)VpK1,…,Kn,Ln≥∏i=1rVKi…,Ki,Kr+1,…,Knp/rVK1,…,Kn-1,Lnp-1.This Lp type inequality has never been discovered before, which is first published and named here. Obviously, for p=1, (17) becomes (8). When r=n, K1=⋯=Kn-1=K, Kn=L, and Ln=K, (16) becomes the following inequality which was established by Gardner, Hug, and Weil [25]. If K is a convex body containing the origin in its interior, L is a convex body containing the origin, and φ∈Φ, then(18)VφK,L≥VK·φVLVK1/n.If φ is strictly convex, equality holds if and only if K and L are homothetic. In Section 5, we show also that (16) in special case yields the following result. If K is a convex body containing the origin in its interior, L is a convex body containing the origin, 0≤i<n, and φ∈Φ, then (see [43])(19)Wφ,iK,L≥WiK·φWiLWiK1/n-i.If φ is strictly convex, equality holds if and only if K and L are homothetic, where Wi(K) is the usual quermassintegral of convex body K.The classical isoperimetric inequality states that ifK is convex body (see, e.g., [50, p.382]), then (20)SKωnn≥VKκnn-1,with equality if and only if K is ball, where S(K) is the surface area of K and κn denotes volume of the unit ball B and its surface area by ωn. The Orlicz isoperimetric inequality is established in Section 5. If K is a convex body containing the origin, φ∈Φ, and 0≤i<n, then(21)VφKn-i-1,Bi+1,KWiK≥φκnWiK1/n-i.If φ is strictly convex, equality holds if and only if K is a ball, where Vφ(K[n-i-1],B[i+1],K) denotes the Orlicz multiple mixed volume Vφ(K,…,K︸n-i-1,B,…,B︸i+1,K). When φ(t)=tp and p≥1, (21) becomes the following Lp-isoperimetric inequality. If K is a convex body containing the origin, p≥1, and 0≤i<n, then (22)nWp,iK,Bωnn-i≥WiKκnn-i-p,with equality if and only if K is a ball, where Wp,i(K,B) denotes the mixed p-quermassintegral of K and B. For i=0 and p=1, (22) becomes isoperimetric inequality (20).In Section6, we establish the following Orlicz-Brunn-Minkowski type inequality. If K1,…,Kn,Ln are convex bodies containing the origin and φ∈Φ, then for ε>0(23)1≥φVK1,…,KnVK1,…,Kn-1,Kn+φ,εLn+ε·φVK1,…,Kn-1,LnVK1,…,Kn-1,Kn+φ,εLn,If φ is strictly convex, equality holds if and only if Kn and Ln are homothetic. Inequality (23) in special case yields the following inequality, which was established by Gardner, Hug, and Weil [25]. If K,L are convex bodies containing the origin and φ∈Φ(24)1≥φVKVK+φL1/n+φVLVK+φL1/n.If φ is strictly convex, equality holds if and only if K and L are homothetic (also see [26]). In fact, the following inequality is also the special case of (23). If K,L are convex bodies containing the origin, φ∈Φ, and 0≤i<n, then (see [43])(25)1≥φWiKWiK+φL1/n-i+φWiLWiK+φL1/n-i.If φ is strictly convex, equality holds if and only if K and L are homothetic.This paper is organized as follows. In Section2, we collect some basic concepts and facts that will be used in the proofs of our results. The Orlicz addition and Orlicz linear combination are introduced in Section 3. In Section 4, we introduce the Orlicz multiple mixed volumes, and some of its basic properties and lemmas are shown and hence Lp-multiple mixed volume Vp(K1,…,Kn,Ln) of convex bodies K1,…,Kn,Ln is also derived. In Section 5, Orlicz-Aleksandrov-Fenchel inequality is established, which in special case yields an Lp-Aleksandrov-Fenchel inequality and Orlicz-Minkowski inequality, respectively. The Orlicz isoperimetric and Urysohn’s inequalities are given, which imply the Lp-isoperimetric inequality and Lp-Urysohn’s inequality, respectively. In Section 6, a new Orlicz-Brunn-Minkowski inequality for the Orlicz addition is established, which is a generalization of the Orlicz-Brunn-Minkowski inequality for volumes and quermassintegrals.


## 2. Notations and Preliminaries

The setting for this paper isn-dimensional Euclidean space Rn. Let Kn denote the set of convex bodies (compact convex subsets with nonempty interiors) in Rn, let Kon be the class of members of Kn containing the origin, and let Koon be those sets in Kn containing the origin in their interiors. We reserve the letter u∈Sn-1 for unit vectors. Sn-1 is the unit sphere. For a compact set K, we write V(K) for the (n-dimensional) Lebesgue measure of K and call this the volume of K. A nonempty closed convex set is uniquely determined by its support function. Support function is homogeneous of degree 1; that is, (26)hK,rx=rhK,x,for all x∈Rn and r≥0. Obviously, for a pair of compact convex sets K and L, we have (27)hK,x≤hL,x,ifandonlyifK⊂L.A function is a support function of a compact convex set if and only if it is positively homogeneous degree one and subadditive. Let d denote the Hausdorff metric on Kn, i.e., for K,L∈Kn, (28)dK,L=hK,u-hL,u∞,where ·∞ denotes the sup-norm on the space of continuous functions C(Sn-1).


### 2.1. Mixed Volumes

IfKi(i=1,2,…,r) are compact convex subset and λi(i=1,2,…,r) are nonnegative real numbers, then fundamental importance is the fact that the volume of ∑i=1rλiKi is a homogeneous polynomial in λi given by (see, e.g., [53]) (29)Vλ1K1+⋯+λnKn=∑i1,…,inλi1…λinVi1…in,where the sum is taken over all n-tuples (i1,…,in) of positive integers not exceeding r. The coefficient Vi1…in depends only on the bodies Ki1,…,Kin and is uniquely determined by (29); it is called the mixed volume of Ki1,…,Kin and is written as V(Ki1,…,Kin). If K1=…=Kn-i=K and Kn-i+1=…=Kn=L, then the mixed volume V(K1,…,Kn) is written as Vi(K,L). If K1=⋯=Kn-i=K, Kn-i+1=⋯=Kn=B, the mixed volume Vi(K,B) is written as Wi(K) and called quermassintegrals (or ith mixed quermassintegrals) of K. If K1=⋯=Kn-i-1=K, Kn-i=⋯=Kn-1=B, and Kn=L, the mixed volume V(K,…,K,B,…,B,L), with i copies of B and (n-i-1) copies of K, is written as Wi(K,L) and called as the mixed quermassintegrals.Aleksandrov [51] (also see Fenchel and Jessen [52], Busemann [54], and Schneider [55]) have shown that, for K∈Kn and i=0,1,…,n-1, there exists a regular Borel measure (ith mixed surface area measure) Si(K,·) on Sn-1, such that the mixed quermassintegrals Wi(K,L) have the following representation: (30)WiK,L=1n-ilimε→0+⁡WiK+εL-WiKε=1n∫Sn-1hL,udSiK,u.A Minkowski inequality for mixed quermassintegrals states that, for K,L∈Kn and 0≤i<n-1, (31)WiK,Ln-i≥WiKn-i-1WiL,with equality if and only if K and L are homothetic (see [54, 56]). A Brunn-Minkowski inequality for mixed quermassintegrals states that, for K,L∈Kn and 0≤i<n-1, (32)WiK+L1/n-i≥WiK1/n-i+WiL1/n-i,with equality if and only if K and L are homothetic (see [2]).


### 2.2.Lp-Mixed Volumes

The best-known inequality concerning volumes of compact convex sets is the Brunn-Minkowski inequality, stating that ifK and L are compact convex sets in Rn, then (33)VK+L1/n≥VK1/n+VL1/n,with equality if and only if K and L are homothetic.The mixed volumeV1(K,L) of compact convex sets K,L is defined by (34)V1K,L≕1nlimε→0+⁡VK+εL-VKε=1n∫Sn-1hL,udSK,u.The Minkowski’s first inequality for K,L states that (35)V1K,L≥VKn-1/nVL1/n,with equality if and only if K and L are homothetic.From the Brunn-Minkowski inequality, using only a little calculus, one can also derive Minkowski’s second inequality (see [57, p.370]) (36)V1K,L2≥VKVK,…,K,L,L,with equality if and only if K and L are homothetic and where K appears (n-2) times.The middle expression in (34) is the first variation of the volume of K with respect to L and the right-hand side of (34) is its integral representation. The Lp-Brunn-Minkowski theory received its greatest single impetus when Lutwak [49] found the appropriate Lp versions of (34) and (35) and their ingredients. By replacing Minkowski addition and scalar multiplication in (34) by Lp addition and its scalar addition and its scalar multiplication, (37)εp·L=ε1/pL,where p≥1. He established the Lp Minkowski mixed volume inequality. (38)VpK,L≥VKn-p/nVLp/n,with equality if and only if K and L are homothetic, where (39)VpK,L≔pnlimε→0+⁡VK+pε·L-VKε,and showed that the Lp-mixed volume has the following integral representation: (40)VpK,L=1n∫Sn-1hL,uphK,u1-pdSK,u.In particular, in the Lp-Brunn-Minkowski theory, S(K,·) is replaced by the p-surface area measure Sp(K,·) given by (41)dSpK,·=hK,u1-pdSK;u.Hence, the integral representation (40) becomes (42)VpK,L=1n∫Sn-1hL,updSpK,u.


### 2.3. The Mixedp-Quermassintegrals

Mixed quermassintegrals are, of course, the first variation of the ordinary quermassintegrals, with respect to Minkowski addition. The mixed quermassintegralsWp,0(K,L),Wp,1(K,L),…,Wp,n-1(K,L) are the first variation of the ordinary quermassintegrals, with respect to Fiery addition: for K,L∈Koon and real p≥1, defined by (see, e.g., [49]) (43)Wp,iK,L=pn-ilimε→0+⁡WiK+pε·L-WiKε.For p≥1 and 0≤i<n and each K∈Koon, there exists a regular Borel measure Sp,i(K;u) on Sn-1, such that the mixed quermassintegral Wp,i(K,L) has the following integral representation: (44)Wp,iK,L=1n∫Sn-1hL,updSp,iK,u,for all L∈Koon. The measure Sp,i(K,·) is absolutely continuous with respect to Si(K,·) and has Radon-Nikodym derivative (45)dSp,iK,·dSiK,·=hK,·1-p,where Si(K,·) is a regular Boel measure on Sn-1. The measure Sn-1(K,·) is independent of the body K and is just ordinary Lebesgue measure, S, on Sn-1. Si(B,·) denotes the i-th surface area measure of the unit ball in Rn. In fact, Si(B,·)=S for all i. The surface area measure S0(K,·) will frequently be written simply as S(K,·). When i=0, Sp,i(K,·) is just the p-surface area measure Sp(K,·) (see [15, 16]). Obviously, putting i=0 in (44), the mixed p-quermassintegrals Wp,i(K,L) become the Lp-mixed volume Vp(K,L).A fundamental inequality for mixedp-quermassintegrals states that (see [49]), for K,L∈Koon, p≥1 and 0≤i<n, (46)Wp,iK,Ln-i≥WiKn-i-pWiLp,with equality if and only if K and L are homothetic. The Lp-Brunn-Minkowski inequality for quermassintegrals was also established. If K,L∈Koon and p≥1 and 0≤i<n, then (47)WiK+pLp/n-i≥WiKp/n-i+WiLp/n-i,with equality if and only if K and L are homothetic.


## 2.1. Mixed Volumes

IfKi(i=1,2,…,r) are compact convex subset and λi(i=1,2,…,r) are nonnegative real numbers, then fundamental importance is the fact that the volume of ∑i=1rλiKi is a homogeneous polynomial in λi given by (see, e.g., [53]) (29)Vλ1K1+⋯+λnKn=∑i1,…,inλi1…λinVi1…in,where the sum is taken over all n-tuples (i1,…,in) of positive integers not exceeding r. The coefficient Vi1…in depends only on the bodies Ki1,…,Kin and is uniquely determined by (29); it is called the mixed volume of Ki1,…,Kin and is written as V(Ki1,…,Kin). If K1=…=Kn-i=K and Kn-i+1=…=Kn=L, then the mixed volume V(K1,…,Kn) is written as Vi(K,L). If K1=⋯=Kn-i=K, Kn-i+1=⋯=Kn=B, the mixed volume Vi(K,B) is written as Wi(K) and called quermassintegrals (or ith mixed quermassintegrals) of K. If K1=⋯=Kn-i-1=K, Kn-i=⋯=Kn-1=B, and Kn=L, the mixed volume V(K,…,K,B,…,B,L), with i copies of B and (n-i-1) copies of K, is written as Wi(K,L) and called as the mixed quermassintegrals.Aleksandrov [51] (also see Fenchel and Jessen [52], Busemann [54], and Schneider [55]) have shown that, for K∈Kn and i=0,1,…,n-1, there exists a regular Borel measure (ith mixed surface area measure) Si(K,·) on Sn-1, such that the mixed quermassintegrals Wi(K,L) have the following representation: (30)WiK,L=1n-ilimε→0+⁡WiK+εL-WiKε=1n∫Sn-1hL,udSiK,u.A Minkowski inequality for mixed quermassintegrals states that, for K,L∈Kn and 0≤i<n-1, (31)WiK,Ln-i≥WiKn-i-1WiL,with equality if and only if K and L are homothetic (see [54, 56]). A Brunn-Minkowski inequality for mixed quermassintegrals states that, for K,L∈Kn and 0≤i<n-1, (32)WiK+L1/n-i≥WiK1/n-i+WiL1/n-i,with equality if and only if K and L are homothetic (see [2]).


## 2.2.Lp-Mixed Volumes

The best-known inequality concerning volumes of compact convex sets is the Brunn-Minkowski inequality, stating that ifK and L are compact convex sets in Rn, then (33)VK+L1/n≥VK1/n+VL1/n,with equality if and only if K and L are homothetic.The mixed volumeV1(K,L) of compact convex sets K,L is defined by (34)V1K,L≕1nlimε→0+⁡VK+εL-VKε=1n∫Sn-1hL,udSK,u.The Minkowski’s first inequality for K,L states that (35)V1K,L≥VKn-1/nVL1/n,with equality if and only if K and L are homothetic.From the Brunn-Minkowski inequality, using only a little calculus, one can also derive Minkowski’s second inequality (see [57, p.370]) (36)V1K,L2≥VKVK,…,K,L,L,with equality if and only if K and L are homothetic and where K appears (n-2) times.The middle expression in (34) is the first variation of the volume of K with respect to L and the right-hand side of (34) is its integral representation. The Lp-Brunn-Minkowski theory received its greatest single impetus when Lutwak [49] found the appropriate Lp versions of (34) and (35) and their ingredients. By replacing Minkowski addition and scalar multiplication in (34) by Lp addition and its scalar addition and its scalar multiplication, (37)εp·L=ε1/pL,where p≥1. He established the Lp Minkowski mixed volume inequality. (38)VpK,L≥VKn-p/nVLp/n,with equality if and only if K and L are homothetic, where (39)VpK,L≔pnlimε→0+⁡VK+pε·L-VKε,and showed that the Lp-mixed volume has the following integral representation: (40)VpK,L=1n∫Sn-1hL,uphK,u1-pdSK,u.In particular, in the Lp-Brunn-Minkowski theory, S(K,·) is replaced by the p-surface area measure Sp(K,·) given by (41)dSpK,·=hK,u1-pdSK;u.Hence, the integral representation (40) becomes (42)VpK,L=1n∫Sn-1hL,updSpK,u.


## 2.3. The Mixedp-Quermassintegrals

Mixed quermassintegrals are, of course, the first variation of the ordinary quermassintegrals, with respect to Minkowski addition. The mixed quermassintegralsWp,0(K,L),Wp,1(K,L),…,Wp,n-1(K,L) are the first variation of the ordinary quermassintegrals, with respect to Fiery addition: for K,L∈Koon and real p≥1, defined by (see, e.g., [49]) (43)Wp,iK,L=pn-ilimε→0+⁡WiK+pε·L-WiKε.For p≥1 and 0≤i<n and each K∈Koon, there exists a regular Borel measure Sp,i(K;u) on Sn-1, such that the mixed quermassintegral Wp,i(K,L) has the following integral representation: (44)Wp,iK,L=1n∫Sn-1hL,updSp,iK,u,for all L∈Koon. The measure Sp,i(K,·) is absolutely continuous with respect to Si(K,·) and has Radon-Nikodym derivative (45)dSp,iK,·dSiK,·=hK,·1-p,where Si(K,·) is a regular Boel measure on Sn-1. The measure Sn-1(K,·) is independent of the body K and is just ordinary Lebesgue measure, S, on Sn-1. Si(B,·) denotes the i-th surface area measure of the unit ball in Rn. In fact, Si(B,·)=S for all i. The surface area measure S0(K,·) will frequently be written simply as S(K,·). When i=0, Sp,i(K,·) is just the p-surface area measure Sp(K,·) (see [15, 16]). Obviously, putting i=0 in (44), the mixed p-quermassintegrals Wp,i(K,L) become the Lp-mixed volume Vp(K,L).A fundamental inequality for mixedp-quermassintegrals states that (see [49]), for K,L∈Koon, p≥1 and 0≤i<n, (46)Wp,iK,Ln-i≥WiKn-i-pWiLp,with equality if and only if K and L are homothetic. The Lp-Brunn-Minkowski inequality for quermassintegrals was also established. If K,L∈Koon and p≥1 and 0≤i<n, then (47)WiK+pLp/n-i≥WiKp/n-i+WiLp/n-i,with equality if and only if K and L are homothetic.


## 3. Orlicz Linear Combination

Throughout the paper, letΦm, m∈N, denote the set of convex functions φ:[0,∞)m→[0,∞) that are strictly increasing in each variable and satisfy φ(0,…,0)=0 and φ(0,…,1,…,0)=1. When m=1, we shall write Φ instead of Φ1.Letm≥2,φ∈Φm, Kj∈Kon, and j=1,…,m; the Orlicz addition of K1,…,Km, denoted by +φ(K1,…,Km), is defined by ([25]) (48)h+φK1,…,Km,x=inf⁡λ>0:φhK1,xλ,…,hKm,xλ≤1,for x∈Rn.Equivalently, the Orlicz radial addition+φ(K1,…,Km) can be defined implicitly (and uniquely) by (49)φhK1,xh+φK1,…,Km,x,…,hKm,xh+φK1,…,Km,x=1,if h(K1,x)+⋯+h(Km,x)>0 and by h(+φ(K1,…,Km),x)=0, if h(K1,x)=⋯=h(Km,x)=0, for all x∈Rn.An important special case is obtained when(50)φx1,…,xm=∑j=1mφxj,for some fixed φ∈Φ such that φ(1)=1, and in this case write +φ(K1,…,Km)=K1+φ⋯+φKm. This means that K1+φ⋯+φKm is defined either by (51)hK1+φ⋯+φKm,x=infλ>0:∑j=1mφhKj,xλ≤1,for all x∈Rn, or by the corresponding special case of (49). From (51), it follows easy that (52)∑j=1mφhKj,xλ=1,if and only if (53)λ=hK1+φ⋯+φKm,x.The Orlicz addition+φ is continuous, monotonic, GL(n) covariant, and projection covariant. The Orlicz linear combination was defined by (see [25]) (54)h+φK1,…,Km,α1,…,αm,x=inf⁡λ>0:∑j=1mαjφhKj,xλ≤1,for all x∈Rn, where K1,…,Km∈Kon, αj≥0, and φ∈Φ. Definition (54) corresponds to taking the function φ in (48) and (49) to be (55)φx1,…,xm=∑j=1mαjφxj,where φ∈Φm.


## 4. Orlicz Multiple Mixed Volumes

Let us introduce Orlicz multiple mixed volume.Definition 1.
Forφ∈Φ, we define Orlicz multiple mixed volume, denoted by Vφ(K1,…,Kn,Ln), defined by (56)VφK1,…,Kn,Ln≕1n∫Sn-1φhKn,uhLn,uhLn,udSK1,…,Kn-1;u,for all K1,…,Kn∈Kon and Ln∈Koon.To derive this Definition1, we need the following lemmas.Lemma 2 ([25]).
IfK,L∈Kon, then (57)K+φ,εL→K in the Hausdorff metric as ε→0+.Lemma 3.
IfK1,…,Kn∈Kon,Ln∈Koon, and φ∈Φ, then for ε>0(58)φl′1·ddεε=0+VK1,…,Kn-1,Ln+φ,εKn=1n∫Sn-1φhKn,uhLn,uhLn,udSK1,…,Kn-1;u.Proof.
Supposeε>0, Kn∈Kon, Ln∈Koon, and u∈Sn-1; let (59)hε=hLn+φ,εKn,u.Since(60)hLn,uhε=φ-11-εφhKn,uhεand from Lemma 2 and noting that φ is continuous function, we obtain (61)ddεε=0+VK1,…,Kn-1,Ln+φ,εKn=1nlimε→0+⁡∫Sn-1hεε·hε-hLn,uhεdSK1,…,Kn-1;u=1nlimε→0+⁡∫Sn-1hε·φhKn,uhε·1-φ-11-εφhKn,u/hε1-1-εφhKn,u/hεdSK1,…,Kn-1;u=1n∫Sn-1hLn,u·φhKn,uhLn,ulimy→1-⁡1-yφ1-φydSK1,…,Kn-1;u=1nφl′1∫Sn-1φhKn,uhLn,uhLn,udSK1,…,Kn-1;u, where (62)y=φ-11-εφhKn,uhε.For any K1,…,Kn∈Kon, Ln∈Koon, and φ∈Φ, we see that either side of (61) is equal to Vφ(K1,…,Kn,Ln), and so this new Orlicz multiple dual mixed volume Vφ(K1,…,Kn,Ln) has been born. Hence, the equation in Definition 1 follows immediately.This theorem plays a central role in our deriving the Orlicz multiple mixed volume. Hence, we give the second kind of different proof.Second Proof.One has(63)dhεdε=ddεhLn,uφ-11-εφhKn,uhε=hLn,udφ-1ydyφhKn,uhε-ε·dφzdzhKn,uhε2dhεdεφ-11-εφhKn,uhε2.where (64)y=1-εφhKn,uhε,and (65)z=hKn,uhε.Hence (66)dhεdε=hLn,udφ-1ydyφhKn,uhεφ-11-εφhKn,uhε2+ε·hKn,uhLn,uhε2dφ-1ydydφzdz.Therefore(67)
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.On the other hand, noting that y→1- as ε→0+, we have(68)dφ-1ydε=limy→1-φ-1y-φ-11y-1=1φl′1.From (64), (66), (67), (68), and Lemma 2 and noticing that φ and φ-1 are continuous functions, (58) yields easy.Lemma 4.
IfK1,…,Kn∈Kon, Ln∈Koon, and φ∈Φ, then (69)VφK1,…,Kn,Ln=φl′1·ddεε=0+VK1,…,Kn-1,Ln+φ,εKn.Proof.
This follows immediately from Definition1 and Lemma 3.Lemma 5 ([25]).
IfKi,Li∈Kon and Ki→K, Li→L as i→∞, then (70)Ki+φ,εLi→K+φ,εL,asi→∞,for ε>0.Let us list some important properties of the Orlicz multiple mixed volume.Lemma 6.
IfK1,…,Kn,Ln,K,L∈Kon, λ1,…,λn≥0, and φ∈Φ, then one has the following:(1)VφK1,…,Kn,Ln≥0.(2)Vφ(K,K,…,K)=V(K).(3) forK⊂L(71)VφK,K2⋯,Kn,Ln≤VφL,K2⋯,Kn,Ln.(4)(72)Vφλ1K1,…,λnKn,λnLn=λ1⋯λnVφK1,…,Kn,Ln.(5) The Orlicz multiple mixed volumeVφ(K1,…,Kn,Ln) is continuous function of K1,…,Kn,Ln.(6)(73)Vφλ1K+λ2L,K2⋯,Kn,Ln=λ1VφK,K2,…,Kn,Ln+λ2VφL,K2,…,Kn,Ln.This shows that the Orlicz multiple mixed volumeVφ(K1,…,Kn,Ln) is linear in its (n-1) variables in front.Proof.
From Definition1, it immediately gives (1), (2), and (3).
Combining Definition1 with the fact (see [58]) (74)Sλ1K1,…,λn-1Kn-1,·=λ1⋯λn-1SK1,…,Kn-1,·,it yields (4) directly.
SupposeLin→Ln, Kij→Kj as i→∞ where j=1,…,n; combining Definition 1 and Lemma 5 with the facts(75)VKi1,…,Kin,Kin+φ,εLin,→VK1,…,Kn,Kn+φ,εLn,and (76)VKi1,…,Kin,Lin,→VK1,…,Kn,Lnas i→∞, it yields (5) directly.
Combining Definition1 and noting that the mixed area measure is Minkowski linear in each of its arguments (see [59]) (77)SK+L,K2⋯,Kn-1,·=SK,K2,…,Kn-1,·+SL,K2⋯,Kn-1,·,it yields (6) directly.Lemma 7 ([25]).
SupposeK,L∈Kon and ε>0. If φ∈Φ, then for A∈GL(n)(78)AK+φ,εL=AK+φ,εAL.We easy find that Orlicz multiple dual mixed volumeVφ(K1,K2,⋯,Kn,Ln) is invariant under simultaneous unimodular centro-affine transformation.Lemma 8.
IfK1,…,Kn,Ln∈Kon and φ∈Φ, then, for A∈SL(n), (79)VφAK1,…,AKn,ALn=VφK1,…,Kn,Ln.Proof.
From (69) and Lemma 7, we have, for A∈SL(n), (80)VφAK1,…,AKn,ALn=φl′1limε→0+⁡VAK1,…,AKn-1,ALn+φ,εAKn-VAK1,…,AKn-1,ALnε=φl′1limε→0+⁡VAK1,…,AKn-1,ALn+φ,εKn-VK1,…,Kn-1,Lnε=φl′1limε→0+⁡VK1,…,Kn-1,Ln+φ,εKn-VK1,…,Kn-1,Lnε=VφK1,…,Kn,Ln.Lemma 9 ([25]).
IfK∈Koon, L∈Kon, and φ∈Φ, then(81)VφK,L=φl′1nlimε→0+⁡VK+φ,εL-VKε.Lemma 10.
IfK∈Koon, L∈Kon, and φ∈Φ, then (82)limε→0+⁡V1K,K+φ,εL-VKε=1nlimε→0+⁡VK+φ,εL-VKε.Proof.
Supposeε>0, K∈Koon, L∈Kon, and u∈Sn-1; let (83)h¯ε=hK+φ,εL,u.From (5), (35), (66), (68), and Lemma 9, we obtain (84)limε→0+⁡V1K,K+φ,εL-VKε=1n∫Sn-1limε→0+⁡h¯ε-hK,uεdSK;u=1n∫Sn-1limε→0+⁡dh¯εdεdSK;u=1nφl′1∫Sn-1φhL,uhK,uhK,udSK;u=1nlimε→0+⁡VK+φ,εL-VKε.Lemma 11.
LetK1,…,Kn∈Kon and Ln∈Koon. If K1=⋯=Kn-1=K, Kn=L, and Ln=K, then (85)VφK1,⋯,Kn,Ln=VφK,L.Proof.
On the one hand, puttingK1=⋯=Kn-1=K, Kn=L, and Ln=K in (69) and noting Lemmas 10 and 9, it follows that (86)VφK1,…,Kn,Ln=φl′1limε→0+⁡V1K,K+φ,εL-VKε=φl′1nlimε→0+⁡VK+φ,εL-VKε=VφK,L.On the other hand, putting K1=⋯=Kn-1=K, Kn=L, and Ln=K in Definition 1 and in view of (5), then (87)VφK1,…,Kn,Ln=1n∫Sn-1φhL,uhK,uhK,udSK1,…,Kn-1;u=VφK,L. Combining (86), (87), and Lemma 3, this shows that(88)VφK1,⋯,Kn,Ln=VφK,L,if K1=⋯=Kn-1=K, Kn=L, and Ln=L.Lemma 12.
IfK∈Koon, L∈Kon and φ∈Φ, and 0≤i<n, then (89)VφKn-i-1,Bi,L,K=1n∫Sn-1φhL,uhK,uhK,udSiK;u.Proof.
On the one hand, puttingK1=⋯=Kn-i-1=K,Kn-i=⋯=Kn-1=B,Ln=K, and Kn=L in (69), from (31), (66), and (68), we obtain for φ∈Φ(90)VφKn-i-1,Bi,L,K=φl′1limε→0+⁡WiK,K+φ,εL-WiKε=1nφl′1∫Sn-1limε→0+⁡hK+φ,εL-hK,uεdSiK;u=1nφl′1∫Sn-1limε→0+⁡dh¯εεdSiK;u=1n∫Sn-1φhL,uhK,uhK,udSiK;u.On the other hand, putting K1=⋯=Kn-i-1=K, Kn-i=⋯=Kn-1=B, Ln=K, and Kn=L, from Definition 1, we have for φ∈Φ(91)VφKn-i-1,Bi,L,K=1n∫Sn-1φhL,uhK,uhK,udSiK;u,where K∈Koon, L∈Kon and φ∈Φ, and 0≤i<n.Here, we denote the Orlicz multiple mixed volumeVφ(K[n-i-1],B[i],L,K) by Wφ,i(K,L) and call Wφ,i(K,L) as Orlicz mixed quermassintegral of convex bodies K and L. When i=0, Orlicz mixed quermassintegral Wφ,i(K,L) becomes Orlicz dual mixed volume Vφ(K,L).Remark 13.
Whenφ(t)=tp, p=1, from the integral representation for Orlicz multiple mixed volume Vφ(K1,…,Kn,Ln) (see Definition 1), it follows easy that Vφ(K1,⋯,Kn,Ln)=V(K1,…,Kn). On the other hand, when φ(t)=tp, p=1, from (69) and noting that φl′(1)=1 and Kn+φ,εLn=Kn+ε·Kn, hence (92)VK1,…,Kn=limε→0+⁡VK1,…,Kn-1,Kn+ε·Kn-VK1,…,Knε. This is very interesting that the mixed volume is such a limiting form.On the other hand, takingLn=Kn in Definition 1 and noting φ(1)=1, Vφ(K1,…,Kn,Ln) also becomes V(K1,…,Kn).Remark 14.
Whenφ(t)=tp and p≥1, we write Vφ(K1,…,Kn,Ln) as Vp(K1,…,Kn,Ln) and call Vp(K1,…,Kn,Ln) as Lp-multiple mixed volume of convex bodies K1,…,Kn,Ln.
Puttingφ(t)=tp, p≥1 in Definition 1, then(93)VpK1,…,Kn,Ln=1n∫Sn-1hLn,u1-phKn,updSK1,…,Kn-1;u.On the other hand, when φ(t)=tp and p≥1, from (69), we get the following a limit of representation of Lp-multiple mixed volume: (94)VpK1,…,Kn,Ln=plimε→0+⁡VK1,…,Kn-1,Ln+pε·Kn,-VK1,…,Kn-1,Lnε.where p≥1 and K1,…,Kn∈Kon, Ln∈Koon.
PuttingK1=⋯=Kn-i-1=K, Kn-i=⋯=Kn-1=B, Kn=L, and Ln=K in (93) and in view of (45) and (46), we obtain(95)VpK1,…,Kn,Ln=1n∫Sn-1hL,uphK,u1-pdSiK,u=1n∫Sn-1hL,updSp,iK,u=Wp,iK,L.Obviously, when Ln=Kn or p=1, from (93), Vp(K1,…,Kn,Ln) becomes the mixed volume V(K1,…,Kn). When K1=⋯=Kn-1=K, Kn=L, and Ln=K, Vp(K1,…,Kn,Ln) becomes Vp(K,L).Lemma 15 (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(96)∫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 [60, p.165]).


## 5. Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Lemma 16.
IfK1,…,Kn∈Kon, Ln∈Koon, and φ∈Φ, then (97)VφK1,…,Kn,LnVK1,…,Kn-1,Ln≥φVK1,…,KnVK1,…,Kn-1,Ln. If φ is strictly convex, equality holds if and only if Kn and Ln are homothetic.Proof.
ForK1,…,Kn∈Kon and any u∈Sn-1, since (98)1nVK1,…,Kn∫Sn-1hKn,udSK1,…,Kn-1;u=1,so h(Kn,u)/nV(K1,…,Kn)S(K1,…,Kn-1;u) is a probability measure on Sn-1.
From (7) and (96) and in view of Definition 1, we obtain (99)VφK1,…,Kn,LnVK1,…,Kn-1,Ln=1nVK1,…,Kn-1,Ln∫Sn-1φhKn,uhLn,uhLn,udSK1,…,Kn-1;u≥φ1nVK1,…,Kn-1,Ln∫Sn-1hKn,udSK1,…,Kn-1;u=φVK1,…,KnVK1,…,Kn-1,Ln.
Next, we discuss the equality condition of (97). Suppose the equality hold in (97), form the equality condition of Jensen’s inequality, it follows that if φ is strictly convex the equality in (97) holds if and only if h(Kn,u)/h(Ln,u) is constant for S(K,…,K;·)-almost all u∈Sn-1. This follows that if φ is strictly convex, the equality in (97) holds if and only if h(Kn,u) and h(Ln,u) proportional for S(K,…,K;·)-almost all u∈Sn-1. This yields that if φ is strictly convex, the equality in (97) holds if and only if Kn and Ln are homothetic.
Conversely, ifφ is strictly convex, then there exist r>0 such that (100)hKn,uhLn,u=r,for S(K,…,K;·)-almost all u∈Sn-1; this shows that(101)r=VK1,…,KnVK1,…,Kn-1,Ln. From Definition 1, (7), and (101), we obtain (102)VφK1,…,Kn,LnVK1,…,Kn-1,Ln=1nVK1,…,Kn-1,Ln∫Sn-1φhKn,uhLn,uhLn,udSK1,…,Kn-1;u=1nVK1,…,Kn-1,Ln×∫Sn-1φVK1,…,KnVK1,…,Kn-1,LnhLn,udSK1,…,Kn-1;u=φVK1,…,KnVK1,…,Kn-1,Ln.This implies that the equality in (97) holds.Theorem 17 (Orlicz-Aleksandrov-Fenchel inequality).
IfK1,…,Kn∈Kon, Ln∈Koon, 1≤r≤n, and φ∈Φ, then (103)VφK1,…,Kn,Ln≥VK1,…,Kn-1,Ln·φ∏i=1rVKi…,Ki,Kr+1,…,Kn1/rVK1,…,Kn-1,Ln.Proof.
This follows immediately from Lemma16 with the Aleksandrov-Fenchel inequality.Unfortunately, precise equality for Orlicz-Aleksandrov-Fenchel inequality is also unknown in general, because the precise equality for the classical Aleksandrov-Fenchel inequality is unknown in general; see [38, p.408] for a full discussion.Corollary 18.
IfK1,…,Kn∈Kon, Ln∈Koon, and φ∈Φ, then (104)VφK1,…,Kn,Ln≥VK1,…,Kn-1,Ln·φVK1⋯VKnVK1,…,Kn-1,Lnn1/n.If φ is strictly convex, equality holds if and only if K1,…,Kn,Ln are all homothetic of each other.Proof.
This inequality (104) yields immediately from (103) with r=n.
Next, we discuss the equality condition of (104). When r=n, the Aleksandrov-Fenchel inequality becomes(105)VK1,…,Knn≥VK1⋯VKn.Although, the precise equality for the Aleksandrov-Fenchel inequality are unknown in general, but it is well known that the equality in (105) holds if and only if K1,…,Kn are all homothetic of each other. Hence combining the equality conditions of Lemma 16, it follows that the equality condition of (104) holds if and only if φ is strictly convex and K1,…,Kn,Ln are all homothetic of each other.Theorem 19.
IfK∈Koon, L∈Kon, 0≤i<n, and φ∈Φ, then(106)Wφ,iK,L≥WiK·φWiLWiK1/n-i.If φ is strictly convex, equality holds if and only if K and L are homothetic.Proof.
This inequality (106) yields immediately from Theorem 17 with r=n-i-1, K1=⋯=Kn-i-1=K,Kn-i=⋯=Kn-1=B, Ln=K, and Kn=L.
Next, we discuss the equality condition of (106). When r=n-i-1, K1=⋯=Kn-i-1=K,Kn-i=⋯=Kn-1=B, Ln=K, and Kn=L, (103) becomes (107)Wφ,iK,L≥WiK·φWiKn-i-1WiL1/n-iWiK.(108)=WiKφWiLWiK1/n-i.When K1=⋯=Kn-i-1=K,Kn-i=⋯=Kn-1=B, Ln=K, and Kn=L, (97) becomes(109)Wφ,iK,L≥WiK·φWiK,LWiK.It is not difficult to see that we derive (107) from (109) by using the well-known inequality and with equality (110)WiK,Ln-i≥WiKn-i-1WiL,with equality if K and L are all homothetic. Although, the precise equality for the Aleksandrov-Fenchel inequality are unknown in general, but combining the equality conditions of (97) and (110), it follows that the equality in (106) holds if φ is strictly convex and if and only if K and L are homothetic.Through the above discussion, it is clear that the equality in (18) (stated in the introduction) holds if and only if K and L are homothetic if φ is strictly convex. Lutwak’s Lp Minkowski inequality (46) follows immediately from (106) with φ(t)=tp and p≥1.There is a quadratic inequality including not only Minkowski first inequality (35), but also Minkowski second inequality (36). This is the celebrated quadratic Aleksandrov-Fenchel inequality, stating that, for compact convex sets K1,…,Kn in Rn,(111)VK1,K2,K3,…,Kn2≥VK1,K1,K3,…,KnVK2,K2,K3,…,Kn.The Aleksandrov-Fenchel inequality appears in the work of Aleksandrov [51] and Fenchel and Jessen [52] and is essentially the most powerful inequality of its type known (see [57, p.371]). Here, we give the following Orlicz quadratic Aleksandrov-Fenchel inequality.Theorem 20 (Orlicz quadratic Aleksandrov-Fenchel inequality).
If K 1 , … , K n ∈ K o n , L n ∈ K o o n,andφ∈Φ, then(112)VφK1,…,Kn,Ln≥VK1,…,Kn-1,Ln·φVK1,K1,K3,…,KnVK2,K2,K3,…,KnVK1,…,Kn-1,Ln21/2.Proof.
This follows immediately from (103).Obviously, whenφ(t)=tp and p=1, (112) becomes (111). On the other hand, putting Ln=Kn and combining ϕ which is increasing function in (112), (112) becomes also (111).Theorem 21 (Orlicz isoperimetric inequality).
IfK∈Kon, φ∈Φ, and 0≤i<n, then (113)VφKn-i-1,Bi+1,KWiK≥φκnWiK1/n-i. If φ is strictly convex, it equality holds if and only if K is a ball.Proof.
This follows immediately from (103) with r=n-i-1, K1=⋯=Kn-i-1=K,Kn-i=⋯=Kn-1=B, Ln=K, and Kn=B.Whenφ(t)=tp, p=1, and i=0, Orlicz isoperimetric inequality (113) becomes the classical isoperimetric inequality. If K is convex body, then (see, e.g., [50, p.382]) (114)SKωnn≥VKκnn-1,with equality if and only if K is ball.The well-known Urysohn’s inequality states that ifK is convex body, then (115)wK2n≥VKκn,with equality if and only if K is ball (see [50, p.382]). Here, w(K) is the mean width of convex body K, denoted by (116)wK≔2κnWn-1K=2κnVK,B,…,B.In the following, we establish an Orlicz Urysohn’s inequality.Theorem 22 (Orlicz Urysohn’s inequality).
IfK∈Kon, φ∈Φ, and 0≤i<n, then (117)VφBn-1,K,Bκn≥φWiKκn1/n-i.If φ is strictly convex, equality holds if and only if K is a ball.Proof.
This follows immediately from (103) with r=n-i-1, K1=⋯=Kn-1=B, Ln=B, and Kn=K.Whenφ(t)=tp and p≥1, the Orlicz Urysohn’s inequality becomes the following Lp-Urysohn’s inequality. If K∈Kon and p≥1 and 0≤i<n, then (118)Wp,iB,Kκnn-i/p≥WiKκn.If φ is strictly convex, equality holds if and only if K is a ball. Putting p=1 and i=0 in (118), (118) becomes the Urysohn’s inequality (115).The following uniqueness is a direct consequence of the Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed volumes.Theorem 23.
IfK1,…,Kn,Ln∈M⊂Kon, and φ∈Φ are strictly convex and if either (119)VφK1,…,Kn-1,Ln,Q=VφK1,…,Kn-1,Kn,Q,forallQ∈M,or(120)VφK1,…,Kn-1,Q,LnVK1,…,Kn-1,Ln=VφK1,…,Kn-1,Q,KnVK1,…,Kn,forallQ∈M,then Kn=Ln.Proof.
Suppose (119) hold. Taking Ln for Q, then from Definition 1 and Lemma 16, we obtain(121)VK1,…,Kn-1,Ln=VφK1,…,Kn,Ln≥VK1,…,Kn-1,Ln·φVK1,…,KnVK1,…,Kn-1,Ln,with equality if and only if Kn and Ln are homothetic. Hence (122)1≥φVK1,…,KnVK1,…,Kn-1,Ln,with equality if and only if Kn and Ln are homothetic. Since φ is increasing function on (0,∞), this follows that (123)VK1,…,Kn≤VK1,…,Kn-1,Ln,with equality if and only if Kn and Ln are homothetic. On the other hand, if taking Kn for Q, we similarly get V(K1,…,Kn)≥V(K1,…,Kn-1,Ln), with equality if and only if Kn and Ln are homothetic. Hence V(K1,…,Kn)=V(K1,…,Kn-1,Ln), and Kn and Ln are homothetic; it follows that Kn and Ln must be equal.
Suppose (120) hold. Taking Kn for Q, then from Definition 1 and Lemma 16, we obtain(124)1=VφK1,…,Kn,LnVK1,…,Kn-1,Ln≥φVK1,…,KnVK1,…,Kn-1,Ln,with equality if and only if Kn and Ln are homothetic. Since φ is increasing function on (0,∞), this follows that (125)VK1,…,Kn-1,Ln≥VK1,…,Kn,with equality if and only if Kn and Ln are homothetic. On the other hand, if taking Ln for Q, we similarly get V(K1,…,Kn-1,Ln)≤V(K1,…,Kn), with equality if and only if Kn and Ln are homothetic. Hence V(K1,…,Kn-1,Ln)=V(K1,…,Kn), and Kn and Ln are homothetic; it follows that Kn and Ln must be equal.Corollary 24.
LetK,L∈M⊂Kon, 0≤i<n, and φ∈Φ be strictly convex, and if either (126)Wφ,iQ,K=Wφ,iQ,L,forallQ∈M,or (127)Wφ,iK,QWiK=Wφ,iL,QWiL,forallQ∈M,then K=L.Proof.
This yields immediately from Theorem23 and Lemma 12.


## 6. Orlicz-Brunn-Minkowski Inequality for Mixed Volumes

Theorem 25 (Orlicz-Brunn-Minkowski inequality).
IfK1,…,Kn,Ln∈Kon, and φ∈Φ, then for ε>0(128)1≥φVK1,…,KnVK1,…,Kn-1,Kn+φ,εLn+ε·φVK1,…,Kn-1,LnVK1,…,Kn-1,Kn+φ,εLn, If φ is strictly convex, equality holds if and only if Kn and Ln are homothetic.Proof.
SupposeKn,Ln∈Kon, and let (129)Q=Kn+φ,εLn.
From Definition1, (7), (48), and (51), and Lemma 16, we obtain (130)VK1,…,Kn-1,Q=1n∫Sn-1φhKn,uhQ,u,hLn,uhQ,uhQ,udSK1,…,Kn-1;u=1n∫Sn-1φhKn,uhQ,u+ε·φhLn,uhQ,uhQ,udSK1,…,Kn-1;u=VφK1,…,Kn,Q+ε·VφK1,…,Kn-1,Ln,Q≥VK1,…,Kn-1,QφVK1,…,KnVK1,…,Kn-1,Q+ε·φVK1,…,Kn-1,LnVK1,…,Kn-1,Q. Taking Q for Kn+φLn in (130), (128) follows.
From the equality condition of Lemma16, the equality in (130) holds if and only if Kn and Q are homothetic, and Ln and Q are homothetic. This yields that if φ is strictly convex, the equality in (130) holds if and only if Kn and Ln are homothetic.Corollary 26.
IfK1,…,Kn,Ln∈Kon, 0≤i≤n, 1≤r≤n, and φ∈Φ, then for ε>0(131)1≥φ∏i=1rVKi,…,Ki,Kr+1,…,Kn1/rVK1,…,Kn-1,Kn+φ,εLn+ε·φ∏i=1,i≠nrVKi,…,Ki,Kr+1,…,Kn-1,Ln1/rVK1,…,Kn-1,Kn+φ,εLn.Proof.
This follows immediately from Theorem25 combining the Aleksandrov-Fenchel inequality.Corollary 27.
IfK1,…,Kn,Ln∈Kon, and φ∈Φ, then for ε>0(132)1≥φVK1⋯VKnVK1,…,Kn-1,Kn+φ,εLnn1/n+ε·VK1⋯VKn-1VLnVK1,…,Kn-1,Kn+φ,εLnn1/n.If φ is strictly convex, equality holds if and only if K1,…,Kn,Ln are all homothetic of each other.Proof.
This inequality (132) follows immediately from (131) with r=n.
Next, we discuss the equality condition of (132). The precise equality for the Aleksandrov-Fenchel inequality is unknown in general, but the following inequality with equality is clear: (133)VK1,…,Knn≥VK1⋯VKn,with equality if and only if K1,…,Kn are all homothetic of each other. Hence combining the equality conditions of (128) and (133), it follows that the equality in (132) holds if φ is strictly convex and if and only if K1,…,Kn,Ln are all homothetic of each other.Corollary 28.
IfK,L∈Kon, φ∈Φ, and 0≤i<n, then for ε>0(134)1≥φWiKWiK+φ,εL1/n-i+ε·φWiLWiK+φ,εL1/n-i. If φ is strictly convex, equality holds if and only if K and L are homothetic.Proof.
This inequality (134) follows immediately from Corollary 26 with r=n-i-1, K1=⋯=Kn-i-1=K+φ,εL  Kn-i=⋯=Kn-1=B, Kn=K, and Ln=L.
Next, we discuss the equality condition of (134). On the one hand, putting r=n-i-1, K1=⋯=Kn-i-1=K+φ,εL  Kn-i=⋯=Kn-1=B, Kn=K, and Ln=L in Theorem 25, we have (135)1≥φWiKWiK+φ,εL1/n-i+ε·φWiLWiK+φ,εL1/n-i.On the other hand, putting K1=⋯=Kn-i-1=K+φ,εL  Kn-i=⋯=Kn-1=B, Kn=K, and Ln=L in Theorem 25, we have(136)1≥φWiK+φ,εL,KWiK+φ,εL+ε·φWiK+φ,εL,LWiK+φ,εL,It is not difficult to see that we derive (135) from (136) by using the following inequalities and with equalities: (137)WiK+φ,εL,Kn-i≥WiK+φ,εLn-i-1WiK,with equality if K and K+φ,εL are all homothetic, and(138)WiK+φ,εL,Ln-i≥WiK+φ,εLn-i-1WiL,with equality if K+φ,εL and L are all homothetic. Although, the precise equality for the Aleksandrov-Fenchel inequality are unknown in general, but combining the equality conditions of (128), (137), and (138), we obtain that the equality in (134) holds if φ is strictly convex, if and only if K and L are homothetic, K and K+φ,εL are homothetic, and K+φ,εL and L are homothetic, this shows that the equality in (134) holds if and only if K and L are homothetic, if φ is strictly convex.Theorem 29 (Lp-Brunn-Minkowski inequality for the mixed volumes).
IfK1,…,Kn,Ln∈Kon, 0≤i<n, 1≤r≤n, and p≥1, then(139)VK1,…,Kn-1,Kn+pLnp≥∏i=1rVKi,…,Ki,Kr+1,…,Knp/r+∏i=1,i≠nrVKi,…,Ki,Kr+1,…,Kn-1,Lnp/r,Proof.
This follows immediately from Corollary26 with φ(t)=tp and p≥1.Puttingr=n-i-1, K1=⋯=Kn-i-1=K+pL, Kn-i=⋯=Kn-1=B, Kn=K, and Ln=L in (139), (139) becomes also Lutwak’s Lp-Brunn-Minkowski inequality (47). Putting r=n in (139), we get the following interesting result: (140)VK1⋯Kn-1,Kn+pLnp≥VK1⋯VKnp/n+VK1⋯VKn-1VLnp/n.with equality if and only if K1,…,Kn,Ln are all homothetic of each other.Corollary 30.
IfK1,…,Kn∈Kon, Ln∈Koon, 1≤r≤n, and φ∈Φ, then (141)VφK1,…,Kn,Ln≥VK1,…,Kn-1,Ln·φ∏i=1rVKi…,Ki,Kr+1,…,Kn1/rVK1,…,Kn-1,Ln.Proof.
Let(142)Kε=Ln+φ,εKn.From (8), (69), and (131), we obtain(143)1φl′1·VφK1,…,Kn,Ln=ddεε=0+VK1,…,Kn-1,Kε=limε→0+⁡VK1,…,Kn-1,Kε-VK1,…,Kn-1,Lnε=limε→0+⁡1-VK1,…,Kn-1,Ln/VK1,…,Kn-1,Kε1-φVK1,…,Kn-1,Ln/VK1,…,Kn-1,Kε·1-φVK1,…,Kn-1,Ln/VK1,…,Kn-1,Kεε·VK1,…,Kn-1,Kε=limt→0+⁡1-tφ1-φ1t·limε→0+⁡1-φVK1,…,Kn-1,Ln/VK1,…,Kn-1,Kεε·limε→0+⁡VK1,…,Kn-1,Kε≥1φl′1·φVK1,…,Kn-1,KnVK1,…,Kn-1,Ln·VK1,…,Kn-1,Ln≥1φl′1·φ∏i=1rKi,…,Ki,Kr+1,…,Kn1/rVK1,…,Kn-1,Ln·VK1,…,Kn-1,Ln. From (143), (141) easy follows. This proof is complete.Finally, it is worth mentioning that we may see that the Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed volumes implies Orlicz-Brunn-Minkowski inequality for the mixed volumes, from the proof of Theorem25. This combines Corollary 30, it yields that the Orlicz-Aleksandrov-Fenchel inequality is equivalent to the Orlicz-Brunn-Minkowski inequality.



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