The Functional Orlicz Brunn-Minkowski Inequality for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 8.58866 12.3952" width="8.58866pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg>-</span>Capacity

Wang, Wei; Li, Juan; He, Rigao; Liu, Lijuan

doi:https://doi.org/10.1155/2020/1670617

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Volume 2020 | Article ID 1670617 | https://doi.org/10.1155/2020/1670617

The Functional Orlicz Brunn-Minkowski Inequality for -Capacity

Wei Wang,¹Juan Li,¹Rigao He,²and Lijuan Liu¹

Academic Editor: Chang Jian Zhao

Received24 Jul 2020

Revised24 Aug 2020

Accepted19 Sept 2020

Published08 Oct 2020

Abstract

In this paper, we establish functional forms of the Orlicz Brunn-Minkowski inequality and the Orlicz-Minkowski inequality for the electrostatic -capacity, which generalize previous results by Zou and Xiong. We also show that these two inequalities are equivalent.

1. Introduction

The classical Brunn-Minkowski inequality was inspired by questions around the isoperimetric problem. It is viewed as one of cornerstones of the Brunn-Minkowski theory, which is a beautiful and powerful tool to conquer all sorts of geometrical problems involving metric quantities such as volume, surface area, and mean width.

An excellent reference on this inequality is provided by Gardner [1].

In 2015, Colesanti, Nyström, Salani, Xiao, Yang, and Zhang (CNSXYZ) [2] introduced the electrostatic -capacity. Let be a compact set in the -dimensional Euclidean space . For , the electrostatic -capacity, , of is defined by where denotes the set of functions from with compact supports and is the characteristic function of . If , then is the classical electrostatic (or Newtonian) capacity of . The Minkowski-type problems for the electrostatic -capacity have attracted increasing attention [2–10]. The electrostatic -capacity also has applications in analysis, mathematical physics, and partial differential equations (see [11–13]).

The electrostatic -capacity can be extended on function spaces. Let denote the set of continuous functions defined on , which is equipped with the metric induced by the maximal norm. Write for the set of strictly positive functions in . For and , define the electrostatic -capacity by where denotes the Aleksandrov body (also known as the Wulff shape) associated with . For nonnegative , the Aleksandrov body is defined by

Obviously, is a compact convex set containing the origin and , where denotes the support function of . Moreover, for every compact convex set containing the origin. If , then is a convex body in containing the origin in its interior. The Aleksandrov convergence lemma reads: if the sequence converges uniformly to , then, .

Suppose (not both zero). If , then, the Minkowski sum is defined by where the scalar multiplication is defined by . By the definition of the Aleksandrov body (3), we have for convex bodies and containing the origin in their interiors. Here, denotes the Minkowski sum of and , i.e., for every , which was defined by Firey [14]. In the mid 1990s, it was shown in [15, 16] that when Minkowski sum is combined with volume the result is an embryonic -Brunn-Minkowski theory. Zou and Xiong ([7], Theorem 3.11) established the functional form of the Brunn-Minkowski inequality for the electrostatic -capacity. Suppose and .

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

The Orlicz Brunn-Minkowski theory which was launched by Lutwak et al. in a series of papers [17–19] is an extension of the Brunn-Minkowski theory. This theory has been considerably developed in the recent years. The Orlicz sum was introduced by Gardner et al. [20]. Let be the class of convex, strictly increasing functions, with . Suppose and (not both zero). If and are convex bodies that contain the origin in their interiors in , then, the Orlicz sum is the convex body defined by for every . Gardner et al. ([20], Corollary 7.5) established the Orlicz Brunn-Minkowski inequality (see also ([21], Theorem 1). Same as the Orlicz sum of convex bodies, we extend the Minkowski sum of functions to the Orlicz sum. For , , and (not both zero), the Orlicz sum is defined by

If we take in (9), then it, induces the Minkowski sum (5). By the definition of the Aleksandrov body (3), (8), (9), and (4), we have for convex bodies and containing the origin in their interiors.

The main aim of this paper is to establish the functional form of the Orlicz Brunn-Minkowski inequality for the electrostatic -capacity.

Theorem 1. Suppose and . If , then, If is strictly convex, equality holds if and only if and are dilates.

2. Notation and Preliminary Results

For excellent references on convex bodies, we recommend the books by Gardner [22], Gruber [23], and Schneider [24].

We will work in equipped with the standard Euclidean norm. Let denote the standard inner product of . For , denotes the Euclidean norm of . We write and for the standard unit ball of and its surface, respectively. Each compact convex set is uniquely determined by its support function , which is defined by , for . Obviously, the support function is positively homogeneous of order 1.

The class of compact convex sets in is often equipped with the Hausdorff metric , which is defined for compact convex sets and by

Denote by the set of convex bodies in and by the set of convex bodies which contain the origin in their interiors. For , the set is called a dilate of convex body . Convex bodies and are said to be homothetic, provided for some and . Let , the Minkowski sum of and is the convex body

Some properties of the electrostatic -capacitary measure are required [2, 3, 7, 8, 11]. The electrostatic -capacitary measure, , of a bounded open convex set in is the measure on the unit sphere defined for and by where (the set of boundary points of ) denotes the inverse Gauss map, the -dimensional Hausdorff measure, and the -equilibrium potential of . If , then the electrostatic q-capacitary measure has the following properties. First, it is positively homogeneous of degree , i.e., for . Second, it is translation invariant, i.e., for . Third, its centroid is at the origin, i.e., . Moreover, it is absolutely continuous with respect to the surface area measure . The weak convergence of the electrostatic -capacitary measure is proved by CNSXYZ ([2], Lemma 4.1): if converges to , then converges weakly to .

CNSXYZ [2] showed the Hadamard variational formula for the electrostatic -capacity: for and ,

And variational formula (14) leads to the following Poincare -capacity formula:

The electrostatic -capacity has the following properties. First, it is increasing with respect to set inclusion; that is, if , then . Second, it is positively homogeneous of degree , i.e., for . Third, it is a rigid motion invariant, i.e.,

for and . If , then (15) induces the Poincare capacity formula

Let denote the set of continuous functions defined on , which is equipped with the metric induced by the maximal norm. Write for the set of strictly positive functions in . Let and . There is a such that for . The Aleksandrov body is continuous in . The Hadamard variational formula for the electrostatic -capacity [2] states the following:

For , define

Obviously, for every . By the Aleksandrov convergence lemma and the continuity of on , the functional is continuous. For and , the mixed electrostatic -capacity is defined by

Applying the Hadamard variational formula for the electrostatic -capacity, the mixed electrostatic -capacity has the following integral representation:

Let . If , then, is the mixed electrostatic -capacity , which has the following integral representation:

The Minkowski inequality for the electrostatic -capacity ([2], Theorem 3.6) states the following: let.

If , then, with equality if and only if and are homothetic.

Let and . For and , the Hadamard variational formula for the electrostatic -capacity [7] states the following:

The mixed electrostatic -capacity is defined by

Take in (24), and combine to obtain the Poincare -capacity formula (15). Zou and Xiong ([7], Theorem 3.9) established the Minkowski inequality for the electrostatic -capacity: let and . If and , then, with equality if and only if and are dilates.

Based on the Orlicz sum (9), we define the Orlicz mixed electrostatic -capacity as follows. For and , the Orlicz mixed electrostatic -capacity is defined by

Indeed, the Orlicz mixed electrostatic -capacity can be extended on function spaces. Let and . For and , the Orlicz mixed electrostatic -capacity is defined by

If with , then, . If with , then, is the Orlicz mixed electrostatic -capacity . In particular, for every .

3. Main Results

The following variational formula of electrostatic -capacity plays a crucial role in our proof.

Lemma 2 ([2], Lemma 5.1). Let be an interval containing 0 in its interior, and let be continuous such that the convergence in is uniformly on . Then, Suppose , , and (not both zero). For every given , the function is strictly decreasing. By the definition of the Orlicz sum (9), we have and only if .
for every .

The continuity properties of the Orlicz sum were established by Xi et al. [21].

Lemma 3 ([21], Lemma 3.1). Suppose,, , and (not both zero). (i)Let and such that and , respectively. Then, (ii)Let such that . Then, (iii)Let (not both zero) such that and . Then

Due to Lemma 2, the integral representation of the Orlicz mixed electrostatic -capacity is given.

Lemma 4. Suppose and . If and , then

Proof. Take an interval for . Denote by Then, the definition of the Orlicz sum (9) and Lemma 3 imply that the function is continuous. By (9), we have for every . Since , we obtain Note that as and the fact that . Thus, uniformly on , where denotes the left derivative of at . Apply Lemma 2 and (35) to get Thus, (27) and (36) yield the desired lemma.

Indeed, (36) can be considered as the Orlicz Hadamard variational formula for the electrostatic -capacity. If we take and with in (36), then, we obtain the Hadamard variational formula (23).

Note that for every . Take in Lemma 4 to get

Lemma 5. Suppose and . If and , then, A direct consequence of Lemma 4 and the homogeneity of the electrostatic q-capacitary measure can be obtained.

Corollary 6. Suppose and . If , then for every .

Let , , and . Note that , and apply (18) and (36) to obtain

Based on (39), one can define the Orlicz mixed electrostatic -capacity of convex bodies K and L as follows: which was first defined by Hong et al. ([10], Definition 3.1).

Lemma 7. Suppose , , , and . (i)Let . If , then (ii)Let and such that and , respectively. Then, (iii)Let such that . Then,

Proof. It follows from (31) that (i) holds if .
Since and , uniformly on ; it follows that uniformly on . Note that , we have uniformly on . The Aleksandrov convergence lemma implies that uniformly on . Meanwhile, the convergence implies that weakly. Applying Lemma 4, one concludes that (ii) holds.
Clearly, there exists a compact interval such that for all .
(iii) directly follows from Lemma 4 and the fact that the sequence converges uniformly to on .
Next, we show that there is a natural Orlicz extension of the Minkowski inequality for the electrostatic -capacity.

Theorem 8. Suppose and . If , then, If is strictly convex, then equality holds if and only if and are dilates.

Proof. By the definition of the mixed electrostatic -capacity (20) and the fact that , we have for every . From (31), Jensen’s inequality, (20), (42), (22), and (18), it follows that It remains to prove the equality condition. Now, suppose is strictly convex. If equality in (41) holds, then, by the equality condition of Jensen’s inequality, there exists an such that for almost every with respect to the measure . Then, we have where the last step is from the equality condition of (42). The definition of Aleksandrov body implies that for almost every with respect to the measure . Thus, for almost every with respect to the measure . By the equality condition of the Minkowski inequality for the electrostatic -capacity, there exists such that .
Hence, for almost every with respect to the measure , Since the centroid of is at the origin, we have that for almost every with respect to the measure . Note that the electrostatic q-capacitary measure is not concentrated on any great subsphere of . Hence, , which in turn implies that and are dilates.
Conversely, assume that and are dilates, say, for some . From our assumption, Corollary 6, (18), and the fact that , it follows that This completes the proof.

By using the Orlicz-Minkowski inequality for the electrostatic -capacity, we establish the following Orlicz Brunn-Minkowski inequality for the electrostatic -capacity which is the general version of Theorem 1.

Theorem 9. Suppose and . If and (not both zero); then, If is strictly convex, then equality holds if and only if and are dilates.

Proof. By (31), (9), and the Orlicz-Minkowski inequality for the electrostatic -capacity (41), we have By the equality condition of the Orlicz-Minkowski inequality for the electrostatic -capacity, we have that if is strictly convex, then equality in (48) holds if and only if and are dilates of .

Remark 1. The case of Theorem 9 was established by Zou and Xiong [7].

For , take and in Theorem 9 to obtain the following Orlicz-Brunn-Minkowski inequality for the electrostatic -capacity, which was established by Hong et al. [10].

Corollary 10 ([10], Theorem 4.2). Suppose and . If, then If is strictly convex, then equality holds if and only if and are dilates.

Remark 2. The case of Corollary 10 was obtained by Colesanti and Salani [25]. Borell [26] first established the Brunn-Minkowski inequality for the classical electrostatic capacity, while its equality condition was shown by Caffarelli et al. [4].

Theorem 11. Suppose , , and . Then, the Orlicz-Brunn-Minkowski inequality for the electrostatic -capacity implies the Orlicz-Minkowski inequality for the electrostatic -capacity.

Proof. For and , define the function by The Orlicz-Brunn-Minkowski inequality for the electrostatic -capacity implies that is nonnegative. Obviously, . Thus, On the other hand, by (51) and the continuity of , we have Let . Note that as . Consequently, The continuity of and (27) imply From (53), (54), (55), and (52), it follows that which implies the Orlicz-Minkowski inequality for the electrostatic -capacity (41).
Finally, we show an immediate application of the Orlicz-Minkowski inequality for the electrostatic -capacity.

Lemma 12. Suppose and . If and is a subset of such that , then the following assertions hold: (i) for all ; then (ii) for all ; then

Proof. We first show that (i) holds. Since , it follows that by the assumption. By the Orlicz-Minkowski inequality for the electrostatic -capacity, we have . The monotonicity of and imply that with equality if and only if and are dilates. This inequality is reversed if interchanging and . So, and and are dilates. Assume that for some . The homogeneity of implies . Thus, .
Then, we can prove (ii) with the similar arguments in (i).

If the Orlicz mixed electrostatic -capacity is restricted on convex bodies, then we obtain the following characterizations for identity of convex bodies, which were proved by Hong et al. [13].

Corollary 13 ([10], Theorem 3.3). Suppose and . If and is a subset of such that , then the following assertions hold: (i) for all ; then (ii)for all ; then

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors have read and approved the final manuscript.

Acknowledgments

L. Liu was supported by the Natural Science Foundation of Hunan Province (No. 2019JJ50172).

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Copyright © 2020 Wei Wang et al. 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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