Journal of Function Spaces
Volume 2018 (2018), Article ID 9752178, 16 pages
https://doi.org/10.1155/2018/9752178

Research Article

Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Chang-Jian Zhao

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

Correspondence should be addressed to Chang-Jian Zhao; chjzhao@163.com

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 [3–22] 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 [31–48].

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 , then

If 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 , then

If 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

Since

and 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 has

where

and

Hence

Therefore

On the other hand, noting that as , we have

From (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 facts

and

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

Lemma 11. Let and . If , , and , then

Proof. On the one hand, putting , , and in (69) and noting Lemmas 10 and 9, it follows that

On the other hand, putting , , and in Definition 1 and in view of (5), then

Combining (86), (87), and Lemma 3, this shows that

if , , and .

Lemma 12. If , and , and , then

Proof. On the one hand, putting , and in (69), from (31), (66), and (68), we obtain for

On the other hand, putting , , , and , from Definition 1, we have for

where , and , and .

Here, we denote the Orlicz multiple mixed volume by and call as Orlicz mixed quermassintegral of convex bodies and . When , Orlicz mixed quermassintegral becomes Orlicz dual mixed volume .

Remark 13. When , , from the integral representation for Orlicz multiple mixed volume (see Definition 1), it follows easy that On the other hand, when , , from (69) and noting that and , hence

This is very interesting that the mixed volume is such a limiting form.

On the other hand, taking in Definition 1 and noting , also becomes

Remark 14. When and , we write as and call as -multiple mixed volume of convex bodies .
Putting , in Definition 1, then

On the other hand, when and , from (69), we get the following a limit of representation of -multiple mixed volume:

where and , .
Putting , , , and in (93) and in view of (45) and (46), we obtain

Obviously, when or , from (93), becomes the mixed volume When , , and , becomes

Lemma 15 (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 [60, p.165]).

5. Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Lemma 16. If , , and , then

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

Proof. For and any , since

so is a probability measure on .
From (7) and (96) and in view of Definition 1, we obtain

Next, we discuss the equality condition of (97). Suppose the equality hold in (97), form the equality condition of Jensen’s inequality, it follows that if is strictly convex the equality in (97) holds if and only if is constant for -almost all . This follows that if is strictly convex, the equality in (97) holds if and only if and proportional for -almost all . This yields that if is strictly convex, the equality in (97) holds if and only if and are homothetic.
Conversely, if is strictly convex, then there exist such that

for -almost all ; this shows that

From Definition 1, (7), and (101), we obtain

This implies that the equality in (97) holds.

Theorem 17 (Orlicz-Aleksandrov-Fenchel inequality). If , , , and , then

Proof. This follows immediately from Lemma 16 with the Aleksandrov-Fenchel inequality.

Unfortunately, precise equality for Orlicz-Aleksandrov-Fenchel inequality is also unknown in general, because the precise equality for the classical Aleksandrov-Fenchel inequality is unknown in general; see [38, p.408] for a full discussion.

Corollary 18. If , , and , then

If is strictly convex, equality holds if and only if are all homothetic of each other.

Proof. This inequality (104) yields immediately from (103) with .
Next, we discuss the equality condition of (104). When , the Aleksandrov-Fenchel inequality becomes

Although, the precise equality for the Aleksandrov-Fenchel inequality are unknown in general, but it is well known that the equality in (105) holds if and only if are all homothetic of each other. Hence combining the equality conditions of Lemma 16, it follows that the equality condition of (104) holds if and only if is strictly convex and are all homothetic of each other.

Theorem 19. If , , , and , then

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

Proof. This inequality (106) yields immediately from Theorem 17 with , , , and
Next, we discuss the equality condition of (106). When , , , and , (103) becomes

When , , and , (97) becomes

It is not difficult to see that we derive (107) from (109) by using the well-known inequality and with equality

with equality if and are all homothetic. Although, the precise equality for the Aleksandrov-Fenchel inequality are unknown in general, but combining the equality conditions of (97) and (110), it follows that the equality in (106) holds if is strictly convex and if and only if and are homothetic.

Through the above discussion, it is clear that the equality in (18) (stated in the introduction) holds if and only if and are homothetic if is strictly convex. Lutwak’s Minkowski inequality (46) follows immediately from (106) with and .

There is a quadratic inequality including not only Minkowski first inequality (35), but also Minkowski second inequality (36). This is the celebrated quadratic Aleksandrov-Fenchel inequality, stating that, for compact convex sets in ,

The Aleksandrov-Fenchel inequality appears in the work of Aleksandrov [51] and Fenchel and Jessen [52] and is essentially the most powerful inequality of its type known (see [57, p.371]). Here, we give the following Orlicz quadratic Aleksandrov-Fenchel inequality.

Theorem 20 (Orlicz quadratic Aleksandrov-Fenchel inequality). If , , and , then

Proof. This follows immediately from (103).

Obviously, when and , (112) becomes (111). On the other hand, putting and combining which is increasing function in (112), (112) becomes also (111).

Theorem 21 (Orlicz isoperimetric inequality). If , , and , then

If is strictly convex, it equality holds if and only if is a ball.

Proof. This follows immediately from (103) with , , , and

When , , and , Orlicz isoperimetric inequality (113) becomes the classical isoperimetric inequality. If is convex body, then (see, e.g., [50, p.382])

with equality if and only if is ball.

The well-known Urysohn’s inequality states that if is convex body, then

with equality if and only if is ball (see [50, p.382]). Here, is the mean width of convex body , denoted by

In the following, we establish an Orlicz Urysohn’s inequality.

Theorem 22 (Orlicz Urysohn’s inequality). If , , and , then

If is strictly convex, equality holds if and only if is a ball.

Proof. This follows immediately from (103) with , , , and

When and , the Orlicz Urysohn’s inequality becomes the following -Urysohn’s inequality. If and and , then

If is strictly convex, equality holds if and only if is a ball. Putting and in (118), (118) becomes the Urysohn’s inequality (115).

The following uniqueness is a direct consequence of the Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed volumes.

Theorem 23. If , and are strictly convex and if either

or

then

Proof. Suppose (119) hold. Taking for , then from Definition 1 and Lemma 16, we obtain

with equality if and only if and are homothetic. Hence

with equality if and only if and are homothetic. Since is increasing function on , this follows that

with equality if and only if and are homothetic. On the other hand, if taking for , we similarly get , with equality if and only if and are homothetic. Hence , and and are homothetic; it follows that and must be equal.
Suppose (120) hold. Taking for , then from Definition 1 and Lemma 16, we obtain

with equality if and only if and are homothetic. Since is increasing function on , this follows that

with equality if and only if and are homothetic. On the other hand, if taking for , we similarly get , with equality if and only if and are homothetic. Hence , and and are homothetic; it follows that and must be equal.

Corollary 24. Let , , and be strictly convex, and if either

or

then

Proof. This yields immediately from Theorem 23 and Lemma 12.

6. Orlicz-Brunn-Minkowski Inequality for Mixed Volumes

Theorem 25 (Orlicz-Brunn-Minkowski inequality). If , and , then for

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

Proof. Suppose , and let

From Definition 1, (7), (48), and (51), and Lemma 16, we obtain

Taking for in (130), (128) follows.
From the equality condition of Lemma 16, the equality in (130) holds if and only if and are homothetic, and and are homothetic. This yields that if is strictly convex, the equality in (130) holds if and only if and are homothetic.

Corollary 26. If , , , and , then for

Proof. This follows immediately from Theorem 25 combining the Aleksandrov-Fenchel inequality.

Corollary 27. If , and , then for

If is strictly convex, equality holds if and only if are all homothetic of each other.

Proof. This inequality (132) follows immediately from (131) with
Next, we discuss the equality condition of (132). The precise equality for the Aleksandrov-Fenchel inequality is unknown in general, but the following inequality with equality is clear:

with equality if and only if are all homothetic of each other. Hence combining the equality conditions of (128) and (133), it follows that the equality in (132) holds if is strictly convex and if and only if are all homothetic of each other.

Corollary 28. If , , and , then for

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

Proof. This inequality (134) follows immediately from Corollary 26 with ,   , , and .
Next, we discuss the equality condition of (134). On the one hand, putting ,   , , and in Theorem 25, we have

On the other hand, putting   , , and in Theorem 25, we have

It is not difficult to see that we derive (135) from (136) by using the following inequalities and with equalities:

with equality if and are all homothetic, and

with equality if and are all homothetic. Although, the precise equality for the Aleksandrov-Fenchel inequality are unknown in general, but combining the equality conditions of (128), (137), and (138), we obtain that the equality in (134) holds if is strictly convex, if and only if and are homothetic, and are homothetic, and and are homothetic, this shows that the equality in (134) holds if and only if and are homothetic, if is strictly convex.

Theorem 29 (-Brunn-Minkowski inequality for the mixed volumes). If , , , and , then

Proof. This follows immediately from Corollary 26 with and .

Putting , , , , and in (139), (139) becomes also Lutwak’s -Brunn-Minkowski inequality (47). Putting in (139), we get the following interesting result:

with equality if and only if are all homothetic of each other.

Corollary 30. If , , , and , then

Proof. Let

From (8), (69), and (131), we obtain

From (143), (141) easy follows. This proof is complete.

Finally, it is worth mentioning that we may see that the Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed volumes implies Orlicz-Brunn-Minkowski inequality for the mixed volumes, from the proof of Theorem 25. This combines Corollary 30, it yields that the Orlicz-Aleksandrov-Fenchel inequality is equivalent to the Orlicz-Brunn-Minkowski inequality.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Authors’ Contributions

Chang-Jian Zhao provided the questions and gave the proof for the all results.

Acknowledgments

The author expresses his thanks to Professors G. Leng and W. Li for their valuable help. The author’s research is supported by National Natural Science Foundation of China (11371334 and 10971205).

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