Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2018, Article ID 9752178, 16 pages
https://doi.org/10.1155/2018/9752178
Research Article

Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang, China

Correspondence should be addressed to Chang-Jian Zhao; moc.361@oahzjhc

Received 4 May 2018; Revised 9 August 2018; Accepted 14 August 2018; Published 16 September 2018

Academic Editor: Alberto Fiorenza

Copyright © 2018 Chang-Jian Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Our main aim is to generalize the classical mixed volume 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 it Orlicz multiple mixed volume of convex bodies , and , denoted by , which involves convex bodies in . 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 -multiple mixed volume are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yields -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.

1. Introduction

One of the most important operations in geometry is vector addition. As an operation between sets and , defined by 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 -Brunn-Minkowski theory. The set, called addition, was introduced by Firey in [1, 2]. The operation, denoted by , is defined by for all , , and compact convex sets and in containing the origin. When , (2) is interpreted as , as is customary. Here the functions are the support functions. If is a nonempty closed (not necessarily bounded) convex set in , then for , defines the support function of . addition, volume, and combine inequalities lead to the -Brunn-Minkowski theory, which are the fundamental and core content in the Brunn-Minkowski theory. For some important results and more information from this theory, we refer to [322] and the references therein.

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 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 -addition, -mixed volume, -affine surface area, -centroid body, and -projection body and inequalities were extended to an Orlicz setting. Advances in the theory and its dual can be found in [3148].

In 2014, Gardner, Hug, and Weil [25] introduced the Orlicz addition of compact convex sets and in containing the origin, implicitly, by where is a convex and increasing function such that and . Let denote the set of convex functions that is increasing and satisfies and . For and , the new addition becomes . The Orlicz mixed volume with respect to the Orlicz addition, denoted by , is defined by where is the surface area measure of , is a convex body containing the origin in its interior, and is a compact convex set containing the origin. For and , becomes , defined by (see [49]) Associated with the convex bodies in is a unique positive Borel measure on , ; call it the mixed area measure of . For any convex body , one has the integral representation (see, e.g., [5], p. 354). The integration is with respect to the mixed area measure on . The mixed area measure is symmetric in its (first ) arguments. When and , the mixed area measure with copies of and copies of will be written as . For , reduces to the surface area measure . The classical Aleksandrov-Fenchel inequality is as follows: if are convex bodies containing the origin and (see, e.g., [50, p. 401]), then

In the paper, we further improve the Orlicz-Brunn-Minkowski theory. Our main aim is to introduce a new mixed volume of convex bodies , denoted as , which involves convex bodies in , and call it Orlicz multiple mixed volume for convex bodies , and . 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 -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 -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 and where are convex bodies containing the origin, is a convex body containing the origin in its interior, is the usual mixed volume, and denotes the value of left derivative of convex function at point 1. Here, denotes the Orlicz linear combination of convex bodies and . If are convex bodies containing the origin, , and , then Orlicz linear combination is defined by ([25]) For and , the Orlicz linear combination is denoted by . This first-order variational equation then provides the new geometric quantity ; it is called it Orlicz multiple mixed volume of convex bodies and defined by We prove also the new affine geometric quantity which has an integral representation.When and , becomes a new mixed volume ; call it -multiple mixed volume, which is first published and named here. For , becomes , and for , , and , becomes the -mixed quermassintegral of and , defined by, for all (see [49]), Putting , , and in (12), becomes the well-know Orlicz mixed volume This shows that the well-known mixed volumes , , and are all the special cases of , where and . In fact, the Orlicz mixed quermassintegral is also a special case of . The Orlicz mixed quermassintegral of and , , is defined by (see Section 4)

In Section 5, we establish the following Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed volume. If are convex bodies containing the origin, is a convex body containing the origin in its interior, , and , then When and , (16) becomes a new -Aleksandrov-Fenchel type inequality. If are convex bodies containing the origin, is a convex body containing the origin in its interior, , and , then This type inequality has never been discovered before, which is first published and named here. Obviously, for , (17) becomes (8). When , , , and , (16) becomes the following inequality which was established by Gardner, Hug, and Weil [25]. If is a convex body containing the origin in its interior, is a convex body containing the origin, and , thenIf is strictly convex, equality holds if and only if and are homothetic. In Section 5, we show also that (16) in special case yields the following result. If is a convex body containing the origin in its interior, is a convex body containing the origin, , and , then (see [43])If is strictly convex, equality holds if and only if and are homothetic, where is the usual quermassintegral of convex body .

The classical isoperimetric inequality states that if is convex body (see, e.g., [50, p.382]), then with equality if and only if is ball, where is the surface area of and denotes volume of the unit ball and its surface area by The Orlicz isoperimetric inequality is established in Section 5. If is a convex body containing the origin, , and , thenIf is strictly convex, equality holds if and only if is a ball, where denotes the Orlicz multiple mixed volume When and , (21) becomes the following -isoperimetric inequality. If is a convex body containing the origin, , and , then with equality if and only if is a ball, where denotes the mixed -quermassintegral of and . For and , (22) becomes isoperimetric inequality (20).

In Section 6, we establish the following Orlicz-Brunn-Minkowski type inequality. If are convex bodies containing the origin and , then for If is strictly convex, equality holds if and only if and are homothetic. Inequality (23) in special case yields the following inequality, which was established by Gardner, Hug, and Weil [25]. If are convex bodies containing the origin and If is strictly convex, equality holds if and only if and are homothetic (also see [26]). In fact, the following inequality is also the special case of (23). If are convex bodies containing the origin, , and , then (see [43])If is strictly convex, equality holds if and only if and are homothetic.

This paper is organized as follows. In Section 2, 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 -multiple mixed volume of convex bodies is also derived. In Section 5, Orlicz-Aleksandrov-Fenchel inequality is established, which in special case yields an -Aleksandrov-Fenchel inequality and Orlicz-Minkowski inequality, respectively. The Orlicz isoperimetric and Urysohn’s inequalities are given, which imply the -isoperimetric inequality and -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 is -dimensional Euclidean space . Let denote the set of convex bodies (compact convex subsets with nonempty interiors) in , let be the class of members of containing the origin, and 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 , we write for the (-dimensional) Lebesgue measure of and call this the volume of . A nonempty closed convex set is uniquely determined by its support function. Support function is homogeneous of degree 1; that is, for all and . Obviously, for a pair of compact convex sets and , we have A function is a support function of a compact convex set if and only if it is positively homogeneous degree one and subadditive. Let denote the Hausdorff metric on , i.e., for , where denotes the sup-norm on the space of continuous functions .

2.1. Mixed Volumes

If are compact convex subset and are nonnegative real numbers, then fundamental importance is the fact that the volume of is a homogeneous polynomial in given by (see, e.g., [53]) where the sum is taken over all -tuples of positive integers not exceeding . The coefficient depends only on the bodies and is uniquely determined by (29); it is called the mixed volume of and is written as If and , then the mixed volume is written as . If , , the mixed volume is written as and called quermassintegrals (or th mixed quermassintegrals) of . If , , and , the mixed volume , with copies of and copies of , is written as and called as the mixed quermassintegrals.

Aleksandrov [51] (also see Fenchel and Jessen [52], Busemann [54], and Schneider [55]) have shown that, for and , there exists a regular Borel measure (th mixed surface area measure) on , such that the mixed quermassintegrals have the following representation: A Minkowski inequality for mixed quermassintegrals states that, for and , with equality if and only if and are homothetic (see [54, 56]). A Brunn-Minkowski inequality for mixed quermassintegrals states that, for and , with equality if and only if and are homothetic (see [2]).

2.2. -Mixed Volumes

The best-known inequality concerning volumes of compact convex sets is the Brunn-Minkowski inequality, stating that if and are compact convex sets in , then with equality if and only if and are homothetic.

The mixed volume of compact convex sets is defined by The Minkowski’s first inequality for states that with equality if and only if and are homothetic.

From the Brunn-Minkowski inequality, using only a little calculus, one can also derive Minkowski’s second inequality (see [57, p.370]) with equality if and only if and are homothetic and where appears times.

The middle expression in (34) is the first variation of the volume of with respect to and the right-hand side of (34) is its integral representation. The -Brunn-Minkowski theory received its greatest single impetus when Lutwak [49] found the appropriate versions of (34) and (35) and their ingredients. By replacing Minkowski addition and scalar multiplication in (34) by addition and its scalar addition and its scalar multiplication, where . He established the Minkowski mixed volume inequality. with equality if and only if and are homothetic, where and showed that the -mixed volume has the following integral representation: In particular, in the -Brunn-Minkowski theory, is replaced by the -surface area measure given by Hence, the integral representation (40) becomes

2.3. The Mixed -Quermassintegrals

Mixed quermassintegrals are, of course, the first variation of the ordinary quermassintegrals, with respect to Minkowski addition. The mixed quermassintegrals are the first variation of the ordinary quermassintegrals, with respect to Fiery addition: for and real , defined by (see, e.g., [49]) For and and each , there exists a regular Borel measure on , such that the mixed quermassintegral has the following integral representation: for all . The measure is absolutely continuous with respect to and has Radon-Nikodym derivative where is a regular Boel measure on . The measure is independent of the body and is just ordinary Lebesgue measure, , on . denotes the -th surface area measure of the unit ball in . In fact, for all . The surface area measure will frequently be written simply as . When , is just the -surface area measure (see [15, 16]). Obviously, putting in (44), the mixed -quermassintegrals become the -mixed volume .

A fundamental inequality for mixed -quermassintegrals states that (see [49]), for , and , with equality if and only if and are homothetic. The -Brunn-Minkowski inequality for quermassintegrals was also established. If and and , then with equality if and only if and are homothetic.

3. Orlicz Linear Combination

Throughout the paper, let , , denote the set of convex functions that are strictly increasing in each variable and satisfy and . When , we shall write instead of .

Let , , and ; the Orlicz addition of , denoted by , is defined by ([25]) for

Equivalently, the Orlicz radial addition can be defined implicitly (and uniquely) by if and by , if , for all .

An important special case is obtained when for some fixed such that , and in this case write This means that is defined either by for all , or by the corresponding special case of (49). From (51), it follows easy that if and only if

The Orlicz addition is continuous, monotonic, covariant, and projection covariant. The Orlicz linear combination was defined by (see [25]) for all , where , , and . Definition (54) corresponds to taking the function in (48) and (49) to be where

4. Orlicz Multiple Mixed Volumes

Let us introduce Orlicz multiple mixed volume.

Definition 1. For , we define Orlicz multiple mixed volume, denoted by , defined by for all and .

To derive this Definition 1, we need the following lemmas.

Lemma 2 ([25]). If , then in the Hausdorff metric as

Lemma 3. If , and , then for

Proof. Suppose , , , and ; let Sinceand from Lemma 2 and noting that is continuous function, we obtain where For any , , and , we see that either side of (61) is equal to , and so this new Orlicz multiple dual mixed volume 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 haswhere and Hence ThereforeOn the other hand, noting that as , we haveFrom (64), (66), (67), (68), and Lemma 2 and noticing that and are continuous functions, (58) yields easy.

Lemma 4. If , , and , then

Proof. This follows immediately from Definition 1 and Lemma 3.

Lemma 5 ([25]). If and , as , then for .

Let us list some important properties of the Orlicz multiple mixed volume.

Lemma 6. If , , and , then one has the following:

(1)

(2)

(3) for

(4)

(5) The Orlicz multiple mixed volume is continuous function of .

(6)

This shows that the Orlicz multiple mixed volume is linear in its () variables in front.

Proof. From Definition 1, it immediately gives (1), (2), and (3).
Combining Definition 1 with the fact (see [58]) it yields (4) directly.
Suppose , as where ; combining Definition 1 and Lemma 5 with the factsand as , it yields (5) directly.
Combining Definition 1 and noting that the mixed area measure is Minkowski linear in each of its arguments (see [59]) it yields (6) directly.

Lemma 7 ([25]). Suppose and . If , then for

We easy find that Orlicz multiple dual mixed volume is invariant under simultaneous unimodular centro-affine transformation.

Lemma 8. If and , then, for ,

Proof. From (69) and Lemma 7, we have, for ,

Lemma 9 ([25]). If , , and , then

Lemma 10. If , , and , then

Proof. Suppose , , , and ; let From (5), (35), (66), (68), and Lemma 9, we obtain