Orlicz log-Minkowski inequality☆
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 which are symmetric with respect to the origin, is the following inequality true? where is the cone-volume measure of L, and is its normalization, and 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 , then for , defines the support function 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 that containing the origin in their interior, then with equality if and only if K and L are homothetic, where is the mixed volume measure , and is its normalization, and denotes the usual mixed volume of L and K, defined by (see [3])
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 are convex bodies in that containing the origin in their interior, , then where denotes the multiple mixed volume probability measure of convex bodies and denotes the mixed surface area measure of .
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 -logarithmic Minkowski inequality, respectively.
The Orlicz log-Minkowski inequality If K and L are convex bodies in that containing the origin in their interior, and is a convex and increasing function such that and , then If φ is strictly convex, equality holds if and only if K are L homothetic, where is the Orlicz mixed volume measure , and is its normalization, and denotes the usual Orlicz mixed volume of L and K, defined by (see [20])
Obviously, when and , (1.4) becomes the following log-Minkowski inequality.
The log-Minkowski inequality If K and L are convex bodies in that containing the origin in their interior, and , then with equality if and only if K and L are homothetic, where is the -mixed volume measure , and is its normalization, and denotes the usual Orlicz mixed volume of L and K, defined by (see [21])
Obviously, when , (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 . Let denote the set of convex bodies (compact convex subsets with nonempty interiors) in , let be those sets in containing the origin in their interiors. We reserve the letter for unit vectors. is the unit sphere. For a compact set K, we write 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 , then the usual mixed volume of L and K, defined by From (3.1), we introduce the mixed volume measure of convex bodies L and K.
Definition 3.1 Mixed volume measure For , the mixed volume measure of L and K, denoted by , defined by
From Definition 3.1, it is not difficult to find the following mixed volume probability measure.
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