Orlicz log-Minkowski inequality

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Abstract

In this paper, the well-known log-Minkowski inequality is extended to the Orlicz space. We first propose and establish an Orlicz logarithmic Minkowski inequality by introducing two new concepts of mixed volume measure and Orlicz mixed volume measure, and using the Orlicz Minkowski inequality for the mixed volumes. The Orlicz logarithmic Minkowski inequality in special case yields the Stancu's logarithmic Minkowski inequality. The Lp-mixed volume measure and Lp-logarithmic Minkowski inequality is first derived here, too.

Introduction

In 2012, Böröczky, Lutwak, Yang and et al. [1] conjecture a logarithmic Minkowski inequality for origin-symmetric convex bodies, the conjectured logarithmic Minkowski inequality was stated the following:

The conjectured logarithmic Minkowski inequality

If K and L are convex bodies in Rn which are symmetric with respect to the origin, is the following inequality true?Sn1ln(hKhL)dVL1nln(V(K)V(L)), where dvL=1nhLdSL is the cone-volume measure of L, and dVL=1V(L)dvL is its normalization, and SL=S(L,) is the surface area measure of L.

The functions are the support functions. If K is a nonempty closed (not necessarily bounded) convex set in Rn, thenhK=max{xy:yK}, for xRn, defines the support function hK of K. A nonempty closed convex set is uniquely determined by its support function.

In 2016, Stancu [2] proved a modified logarithmic Minkowski inequality for non-symmetric convex bodies not symmetric with respect to the origin. The logarithmic Minkowski inequality was given in the following result.

The logarithmic Minkowski inequality

If K and L are convex bodies in Rn that containing the origin in their interior, thenSn1ln(hKhL)dv11nln(V(K)V(L)), with equality if and only if K and L are homothetic, where dv1 is the mixed volume measure dv1=1nhKdSL, and dv¯1=1V1(L,K)dv1 is its normalization, and V1(L,K) denotes the usual mixed volume of L and K, defined by (see [3])V1(L,K)=1nSn1hKdSL.

Recently, this logarithmic Minkowski inequality and the conjectured logarithmic Minkowski inequality have attracted extensive attention and research. The recent research on the logarithmic Minkowski and its dual type inequalities can be found in the references [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. In particularly, the log-Aleksandrov–Fenchel inequality has recently been established in [18] as follows:

The log-Aleksandrov–Fenchel inequality

If L1,,Ln,Kn are convex bodies in Rn that containing the origin in their interior, 1rn, thenSn1ln(hKnhLn)dV(L1,,Ln)ln(i=1rV(Li,,Li,Lr+1,,Ln)1/rV(L1,,Ln1,Kn)), where dV(L1,,Ln)=1nV(L1,,Ln)hLndSL1Ln1 denotes the multiple mixed volume probability measure of convex bodies L1,,Ln and SL1Ln1=S(L1,,Ln1,) denotes the mixed surface area measure of L1,,Ln1.

Obviously, (1.2) is a special case of (1.3). Moreover, the dual form of (1.3) has also been established in [19]. In this paper, our main purpose is to generalize Stancu's logarithmic Minkowski inequality (1.2) to the Orlicz space. In the Section 3, we establish the following Orlicz log-Minkowski inequality by introducing the concepts of mixed volume measure and Orlicz mixed volume measure and using the well-known Orlicz Minkowski inequality. The new inequality which in special cases yield logarithmic Minkowski inequality (1.2) and a new Lp-logarithmic Minkowski inequality, respectively.

The Orlicz log-Minkowski inequality

If K and L are convex bodies in Rn that containing the origin in their interior, and φ:[0,)(0,) is a convex and increasing function such that φ(0)=0 and φ(1)=1, thenSn1ln(φ(hKhL))dvφln(φ((V(K)V(L))1/n)). If φ is strictly convex, equality holds if and only if K are L homothetic, where dvφ is the Orlicz mixed volume measure dvφ=1nφ(hKhL)hLdSL, and dv¯φ=1Vφ(L,K)dvφ is its normalization, and Vφ(L,K) denotes the usual Orlicz mixed volume of L and K, defined by (see [20])Vφ(L,K)=1nSn1φ(hKhL)hLdSL.

Obviously, when φ(x)=xp and p1, (1.4) becomes the following Lp log-Minkowski inequality.

The Lp log-Minkowski inequality

If K and L are convex bodies in Rn that containing the origin in their interior, and p1, thenSn1ln(hKhL)dvpln(V(K)V(L))1/n, with equality if and only if K and L are homothetic, where dvφ is the Lp-mixed volume measure dvp=1nhKphL1pdSL, and dv¯p=1Vp(L,K)dvp is its normalization, and Vp(L,K) denotes the usual Orlicz mixed volume of L and K, defined by (see [21])Vp(L,K)=1nSn1hKphL1pdSL.

Obviously, when p=1, (1.5) becomes the log-Minkowski inequality (1.2).

It is worth mentioning to generalize the convex geometry theory to the Orlicz space, which is a new geometric research direction rising in the last ten years. In recent years, this research direction has attracted the widespread attention of many experts. A number of valuable results are presented (see: [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]).

Section snippets

Notations and preliminaries

The setting for this paper is n-dimensional Euclidean space Rn. Let Kn denote the set of convex bodies (compact convex subsets with nonempty interiors) in Rn, let Kon be those sets in Kn containing the origin in their interiors. We reserve the letter uSn1 for unit vectors. Sn1 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. Let d

Orlicz log-Minkowski inequality

In the section, in order to prove the Orlicz log-Minkowski inequality, we need to define some new mixed volume measures.

If K,LKon, then the usual mixed volume of L and K, defined byV1(L,K)=1nSn1hKdSL. From (3.1), we introduce the mixed volume measure of convex bodies L and K.

Definition 3.1 Mixed volume measure

For L,KKon, the mixed volume measure of L and K, denoted by dv1(L,K), defined bydv1(L,K)=1nhKdSL.

From Definition 3.1, it is not difficult to find the following mixed volume probability measure.dv1(L,K)=1V1(L,K)dv1(L,K).

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    Research is supported by National Natural Science Foundation of China (11371334, 10971205).

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