Orlicz Mean Dual Affine Quermassintegrals

Zhao, Chang-Jian; Cheung, Wing-Sum

doi:https://doi.org/10.1155/2018/8123924

Journal of Function Spaces

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Abstract Introduction Preliminaries Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 8123924 | https://doi.org/10.1155/2018/8123924

Orlicz Mean Dual Affine Quermassintegrals

Chang-Jian Zhao¹and Wing-Sum Cheung²

Academic Editor: Mitsuru Sugimoto

Received23 Sept 2017

Revised12 Dec 2017

Accepted18 Dec 2017

Published01 Feb 2018

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 the -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.

1. Introduction

The radial addition of star sets (compact sets that are star-shaped at and contain ) and can be defined by where if , and are collinear and , otherwise, or by where denotes the radial function of star set , which is defined by for , where is the surface of the unit sphere. Hints as to the origins of the radial addition can be found in [1, p. 235]. If is positive and continuous, will be called a star body. Let denote the set of star bodies about the origin in . 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 , , and , the -radial addition is defined by (see [14]) The -harmonic radial combination for star bodies was introduced: If , , and , then the -harmonic radial addition defined by Lutwak [8] isFor convex bodies, the -harmonic addition was first investigated by Firey [15].

If is a nonempty closed (not necessarily bounded) convex set in , then for , which defined the support function of . A nonempty closed convex set is uniquely determined by its support function. -addition and inequalities are the fundamental and core content in the -Brunn-Minkowski theory. In recent years, a new extension of -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 -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 the -Brunn-Minkowski theory is extended to the Orlicz Brunn-Minkowski theory, it has recently turned to a study extending from -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 of two star bodies and , defined by where , is a convex and decreasing function such that , , and Let denote the class of the convex and decreasing functions . When and , the Orlicz harmonic addition becomes the -harmonic radial addition . The dual Orlicz mixed volume with respect to Orlicz harmonic radial addition, denoted by , is defined by where is the Orlicz linear combination of and , denotes the surface area measure of the unit sphere , and denotes the value of the right derivative of convex function at point .

The dual affine quermassintegrals were defined, for a convex body , by letting , and for (see, e.g., [30], p. 515) where denotes the Grassmann manifold of -dimensional subspaces in , denotes the gauge Haar measure on , denotes the -dimensional volume of intersection of on -dimensional subspace , and denotes the volume of -dimensional unit ball. Gardner [31] showed the Brunn-Minkowski inequality for the dual affine quermassintegrals. If and , then with equality if and only if is a dilate of , 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) for a convex body and and by letting and . Here, denotes the space of the -dimensional affine subspace in and denotes the gauge Haar measure on . They are related to the dual affine quermassintegrals by (see [32], p. 373). Obviously, is invariant under unimodular affine transformations of .

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 the -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 -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 , , , and , Putting in (13), then we have the well-known result. In (13), we find a new geometric quantity. Based on this, we extract the required geometric quantity, denoted by , and call it as Orlicz mean dual affine quermassintegrals, defined by where , , and . We also prove that the new affine geometric quantity has an integral representation. where denotes the Orlicz dual mixed volume of -dimensional star bodies and in -dimensional subspace

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 Section 4, we give a positive answer to this question and establish the dual Orlicz-Minkowski inequality for the new affine geometric quantity. For , , and , we prove the Orlicz-Minkowski inequality for the Orlicz mean dual affine quermassintegrals. If is strictly convex, equality holds if and only if and are dilates. For , (17) becomes the following dual Orlicz-Minkowski inequality established by Zhu et al. [29]: If is strictly convex, equality holds if and only if and are dilates.

In Section 5, 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 , , and , then for any If is strictly convex, equality holds if and only if and are dilates. For and , (19) becomes the following dual Orlicz-Brunn-Minkowski inequality established by Zhu et al. [29]. If and , then If is strictly convex, equality holds if and only if and are dilates. Moreover, for , , and , (19) becomes the -dual Brunn-Minkowski inequality for the mean dual affine quermassintegrals. If , , , , and , then with equality if and only if and are dilates. When , (21) becomes Lutwak’s dual Brunn-Minkowski inequality (36).

2. Preliminaries

The setting for this paper is -dimensional Euclidean space . A body in is a compact set equal to the closure of its interior. For a compact set , we write for the (-dimensional) Lebesgue measure of and call this the volume of . Associated with a compact subset of , which is star-shaped with respect to the origin and contains the origin, its radial function is , defined by Note that the class (star sets) is closed under unions, intersection, and intersection with subspace. The radial function is homogeneous of degree ; that is, , for all and . Let denote the radial Hausdorff metric, as follows; if , then (see, e.g., [30]) From the definition of the radial function, it follows immediately that for the radial function of the image of is given by for all .

2.1. Dual Mixed Volumes and -Dual Mixed Volumes

If , the dual mixed volume defined by (see [11]) is as follows: If , the dual mixed volume is written as . If , the dual mixed volume is written as . Obviously, For , we have and (see [11]) The fundamental inequality for dual mixed volumes stated that if , then with equality if and only if and are dilates. The Brunn-Minkowski inequality for the radial addition is the following: If , then with equality if and only if and are dilates.

The following result follows immediately from the definition of -radial addition, with . Let and ; we define -dual mixed volume of star bodies and , , by This integral representation (30), together with the Hölder inequality, yields the -dual Minkowski inequality (see [36]): If and , then with equality if and only if and are dilates. The definition of -radial addition, together with (31), yields Gardner’s Brunn-Minkowski inequality for -radial addition (see [37]). If and , then with equality if and only if and are dilates.

2.2. -Harmonic Mixed Volumes

The following result follows immediately form (5) with . Let and ; the -harmonic mixed volumes of star bodies and denotes , defined by (see [35]) This integral representation (34), together with the Hölder inequality, yields Lutwak’s -dual Minkowski inequality as follows: If and , then with equality if and only if and are dilates. This integral representation (34), together with the definition of -harmonic addition, yields Lutwak’s -Brunn-Minkowski inequality for harmonic -addition (see [35]). If and , then with equality if and only if and are dilates.

2.3. Orlicz Harmonic Addition and Orlicz Harmonic Linear Combination

Definition 1. Let , , , define the Orlicz harmonic addition of , denoted by , defined by for all .
Equivalently, the Orlicz harmonic addition can be defined implicitly by for all .
The Orlicz harmonic linear combination on the case is defined.

Definition 2. Orlicz harmonic linear combination for , , and (both not zero) is defined by for all .
When and , then Orlicz harmonic linear combination changes to the -harmonic linear combination (see [9]). Moreover, we shall write instead of , for , and assume throughout that this is defined by (39), where , and , and write as .

3. Orlicz Mean Dual Affine Quermassintegrals

In order to define Orlicz mean dual affine quermassintegrals, we need the following lemmas.

Lemma 3. If and , then for

Proof. From (8) and (39), we have for any Putting in (41), (41) easily becomes (40).

Lemma 4 (see [29]). If and , then for in the radial Hausdorff metric as .

Lemma 5. If , , and , then

Proof. Suppose and . For and , we have Hence On the other hand Therefore and are the same star body in

Definition 6. If , , and , then Orlicz mean dual affine quermassintegral of and , denoted by , is defined by Specifically, we agreed on the following: 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. If , , and , then for any

Proof. On the one hand, from (8), we have On the other hand, from (11), (47), and (50), we obtain

Lemma 8. If , , and , then

Proof. The definition of the Orlicz mean dual affine quermassintegrals, together with (8) and (11), gives (52).
If , then and call the -dual mixed mean affine quermassintegral of and , and where denotes the -dual mixed volume of -dimensional star bodies and in -dimensional subspace

Lemma 9 (see [29]). If , , and any , then for

In the following, we will prove that Orlicz mean dual affine quermassintegral is invariant under simultaneous unimodular centro-affine transformation.

Lemma 10. If , , and any , then

Proof. Suppose that and . For any , , and , we have When , let From (23), we obtain On the other hand, from Definition 6 and (58), we have

Next, we also can give another proof directly.

Proof. From Lemmas 7 and 9, we have, for ,

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 , , , and ,

We also need the following lemma to prove our main results.

Lemma 11 (Jensen’s inequality). Let be a probability measure on a space and is a -integrable function, where is a possibly infinite interval. If is a convex function, then If is strictly convex, equality holds if and only if is constant for -almost all (see [38, p.165]).

4. Orlicz-Minkowski Inequality for Orlicz Mean Dual Quermassintegrals

Theorem 12. If , , and , then If is strictly convex, equality holds if and only if and are dilates.

Proof. When , (63) becomes the dual Orlicz-Minkowski inequality; hence we assume . Since the above equation defines a Borel probability measure on ; namely, From (11), (47), and (65) and using dual Orlicz-Minkowski inequality, Jensen inequality, and Hölder inequality, we obtain Next, we discuss the equal condition of (63). If is strictly convex, suppose that and are dilates; that is, there exists such that . Hence 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, Form the equality condition of dual Orlicz-Minkowski inequality, if the equality holds, then and must be dilates. The second inequality in the above proof is Jensen inequality. From the equality condition of Jensen inequality, if is strictly convex and the equality holds, then must be a constant; this yields that and 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 and must be proportional; namely, and are dilates. From the combinations of these equal conditions, it follows that equality in (63) holds, if is strictly convex, and equality holds if and only if and are dilates.

Corollary 13. If , , and , then with equality if and only if and are dilates.

Proof. This follows immediately from (63) with and

Corollary 14. If and , then If is strictly convex, equality holds if and only if and are dilates.

Proof. This follows immediately from (63) with

The following uniqueness is a direct consequence of the Orlicz-Minkowski inequality for the Orlicz mean dual affine quermassintegrals.

Theorem 15. If and is strictly convex, and such that . If or then .

Proof. Suppose that (72) holds. Taking for , then, from Lemma 8 and (63), we obtain with equality if and only if and are dilates. Hence with equality if and only if and are dilates. Since is decreasing function on , it follows that with equality if and only if and are dilates. On the other hand, if taking for , we similarly get , with equality if and only if and are dilates. Hence , and and are dilates; it follows that and must be equal.
Suppose that (73) holds. Taking for , then, from Lemma 8 and (63), we obtain with equality if and only if and are dilates. Hence with equality if and only if and are dilates. Since is decreasing function on , it follows that with equality if and only if and are dilates. On the other hand, if we take for , we similarly get , with equality if and only if and are dilates. Hence , and and are dilates; it follows that and must be equal.

Corollary 16 (see [29]). If and is strictly convex, and such that . If or then .

Proof. This follows immediately from Theorem 15 with

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

Lemma 17. If , , and , then for any

Proof. From (8) and Lemmas 3 and 5, we have Let ; from (83) and (47), we have The proof is complete.

Lemma 18. Let , , and .(1)If and are dilates, then and are dilates.(2)If and are dilates, then and are dilates.

Proof. Suppose that there exists a constant such that ; we have On the other hand, there exists a unique constant such that where satisfies that This shows that
Suppose that there exists a constant such that Then This shows that is a constant. This yields that and are dilates. Namely, and are dilates.

Theorem 19. If , , , and , then If is strictly convex, equality holds if and only if and are dilates.

Proof. From Lemma 17 and (63), we obtain If is strictly convex, from equality condition of the Orlicz-Minkowski inequality, the equality holds if and only if and are dilates, and and are dilates and, combining with Lemma 18, this yields that if is strictly convex, equality holds in (90) if and only if and are dilates.

Corollary 20. If , , , and , then with equality if and only if and are dilates.

Proof. This follows immediately from (90) with and

For and , (92) becomes Lutwak’s dual Brunn-Minkowski inequality (36).

Corollary 21. If and , then If is strictly convex, equality holds if and only if and are dilates.

Proof. This follows immediately from (90) with and .

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, if , , and , then If is strictly convex, equality holds if and only if and are dilates.

Proof.
: Let From Lemmas 4 and 7 and using the Orlicz-Brunn-Minkowski inequality (90), we obtain : 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. If and , then If is strictly convex, equality holds if and only if and are dilates.

Proof. This follows immediately from Theorem 22 with and

Corollary 24. If , , and , then If is strictly convex, equality holds if and only if and are dilates.

Proof. This follows immediately from Theorem 22 with and .

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Chang-Jian Zhao and Wing-Sum Cheung provided the questions and gave the proof for the main results. They all read and approved the manuscript.

Acknowledgments

The first author’s research is supported by Natural Science Foundation of China (11371334). Research of the second author is supported by a HKU Seed Grant for Basic Research. The first author expresses his gratitude to Professor G. Leng for his valuable help.

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Copyright © 2018 Chang-Jian Zhao and Wing-Sum Cheung. 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.

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