Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 2, April 2022
Page(s) 115 - 124
DOI https://doi.org/10.1051/wujns/2022272115
Published online 20 May 2022

© Wuhan University 2022

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Let be the set of convex bodies (compact convex sets with nonempty interior) in the n-dimensional Euclidean space R n . For , their Minkowski sum , and the i-th quermassintegral of K will be denoted by for each . The classical Brunn-Minkowski in- equality for quermassintegrals states that for and , then(1)with equality if and only if K and L are homothetic. The case i=0 of (1) is the classical Brunn-Minkowski inequality (see Ref. [1]). It is the core of the Brunn-Minkowski theory, which is derived from questions around the isoperimetric problem. In Gardner’s excellent survey[1], he summarized the history of this inequality and some applications in other related fields such as elliptic partial differential equations and algebraic geometry. In addition, this inequality helped make a great difference in studying inequalities and witnessed a rapid growth.

In the early 1960s, Firey [2] introduced the Lp -addition. Let be the set of all convex bodies in containing the origin in their interiors. For R n , and , the Lp -Minkowski addition +p is defined by (see Ref. [2])where hK denotes the support function of the convex body K and it is defined by Here x, y denote the standard inner product of x, . Thirty years after the new Lp -addition, Lutwak[3,4] established the Lp -Brunn-Minkowski inequality for quermassintegrals: For , and n-1, thenwith equality if and only if K and L are dilates. Readers can refer to Refs. [5-9] for additional references.

The Orlicz-Brunn-Minkowski theory originated from the work of Lutwak et al in 2010[10,11]. As an important part of the theory, the Orlicz Brunn-Minkowski inequality has been very popular with scholars in related fields. At first, the Orlicz Busemann-Petty centroid inequality[10] was introduced as a new proof by Li and Leng[12] in 2010 and the Orlicz Petty projection inequality were established by Lutwak et al [11]. After that, Gardner et al [13] introduced the Orlicz addition and established the new Orlicz- Brunn-Minkowski inequality that implied the Lp -Brunn-Minkowski inequality in 2014. Let be the class of convex and strictly increasing functions, such that and . For and , the Orlicz addition (see Section 1 for precise definition) is defined by(2)for . In the same year, Xiong and Zou [14] established the Orlicz-Brunn-Minkowski inequality for quermassintegrals: For , and , then(3)If ϕ is strictly convex, equality holds if and only if K and L are dilates. The case i=0 was established by Refs. [13, 15].

In 1975, Lutwak [16] introduced dual mixed volumes and radial addition, and studied the dual Brunn-Minkowski theory for star bodies. In 2015, Gardner et al [17] established the dual Orlicz-Brunn-Minkowski theory and introduced the concept of radial Orlicz addtion. Let be the set of continuous and strictly increasing functions, such that and . Let be the set of continuous and strictly decreasing functions, such that and . Let be the set of all star bodies with the origin as an interior point. For and , the radial Orlicz addtion (see Section 1 for precise definition) is defined by(4)for .

The inequalities for polar bodies and dual star bodies began to attract attention. For instance, Zhu[18] confirmed the conjecture[10] that the Orlicz centroid inequality for convex bodies can be extended to star bodies; Cifre and Nicol[19] proved a Brunn-Minkowski-type in- equality for the polar set of the p-sum of convex bodies, which generalized previous results by Firey[20]; Wang and Huang [21] gave a systematic explanation of Orlicz Brunn-Minkowski inequality for polar bodies and dual star bodies and Liu [22] established some Orlicz-Brunn- Minkowski type inequalities for (dual) quermassintegrals of polar bodies and star dual bodies. Besides, the Orlicz-Brunn-Minkowski inequality for complex projection bodies[23] is also a very active field. For other generalizations on Orlicz spaces, see Refs. [17, 24, 25].

Let be the polar body of a convex body the dual star body of a convex body K. Liu[22] established the following Orlicz-Brunn-Minkowski type inequality for dual quermassintegrals of polar bodies and star dual bodies: For and , then(5)If ϕ is strictly convex, equality holds if and only if K and L are dilates.

For and , if is concave, then(6)while if is convex, the inequality is reversed. If is strictly concave, equality holds if and only if K and L are dilates.

The purpose of this paper is to establish the following Orlicz-Brunn-Minkowski type inequality for dual mixed quermassintegrals of polar bodies and star dual bodies.

Theorem 1   Let and . Then

If ϕ is strictly convex, equality holds if and only if K and L are dilates.

Theorem 2   Let , and . If is concave, then

If is convex, the inequality is reversed. If is strictly concave (or convex, as appropriate), equality holds if and only if K and L are dilates.

Liu[22] also established the following dual Orlicz- Brunn-Minkowski type inequality dual quermassintegrals of polar bodies: Let such that is strictly convex, and , then(7)with equality if and only if K and L are dilates.

We also establish the following dual Orlicz-Brunn -Minkowski type inequality for dual mixed quermassintegrals of polar bodies which is the dual form of Theorem 1.

Theorem 3   Let such that is strictly convex, and , thenwith equality if and only if K and L are dilates.

This paper is organized as follows. In Section 1, we collect some concepts and facts to be used in the proofs of our results. In Section 2, we give the integral forms of some (dual) mixed Orlicz quermassintegrals and confirm that a special case of the Aleksandrov-Fenchel inequality (with respect to three convex bodies) can be generalized to Orlicz setting. In Section 3, we introduce some Orlicz-Brunn-Minkowski inequalities for (dual) mixed quermassintegrals of polar bodies and star dual bodies.

1 Preliminaries

For , we define the Orlicz sum by (see Ref. [13])for .

Equivalently, the Orlicz sum can be defined implicitly byIf , and by , if . Here , the set of convex functions that are increasing in each variable with , and . In particular, if , and p≥1, then Orlicz addition reduces to Lp addition.

Gardner, Hug and Weil [13] proved that Orlicz addition is commutative if and only if . For some , the set of convex functions satisfy , and . Therefore, (2) was defined.

For a compact star-shaped set K about the origin, the radial function is defined bywhere the compact star-shaped set is defined if the intersection of every straight line through the origin with K is a line segment. And if the is positive and continuous, then the compact star-shaped set K about the origin is called a star body.

For , the radial addition is defined by(8)and if s>0, then for all ,(9)For , we define the radial Orlicz sum by (see Ref.[17])for .

Equivalently, the radial Orlicz addition can be defined implicitly byIf , and by , if . An important special case is obtained when , for fixed .

Then by the corresponding special case(10)when , and by , otherwise, and similarly by (10) when .

Therefore, (4) was defined.

We denote the unit ball in R n and its surface by , respectively. The dual mixed volume is defined by (see Ref. [16])(11)where S is the spherical Lebesgue measure ((n−1) dimensional Hausdorff measure) of S n-1.

The polar body K * of a convex body K is defined byand it is easy to see that K* is a convex body and . If (a convex body that contains the origin in its interior), for all ,(12)

Suppose that μ is a probability measure on a space X and is a μ-integrable function, where I is a possible infinite interval. Jensen’s inequality states that if is a convex function, then(13)

When ϕ is strictly convex, equality holds if and only if g(x) is a constant for μ-almost all (see Refs. [22, 26]). If ϕ is a concave function, the inequality is reversed.

For a convex body K, the i-th quermassintegral of K, , has the following integral representation:where is -th surface area measures of K. In particular, , , and , where B is the unit ball in R n , and V, S denote the volume and the surface area of the set involved, respectively.

For , the mixed quermassintegral has the following integral representation:In particular, .

For , , the mixed volume is written as . In particular, , . The mixed quermassintegral has the following integral representation (see Ref. [9]):(14)where the measure

An important special case of the Aleksandrov-Fenchel inequality [9] is stated as follows:

Suppose , then for ,(15)and the inequality can be rewritten as(16)We will extend the inequality (16) to the Orlicz setting in Theorem 4. Clearly, the equality holds in (15) and (16) if K and L are homothetic. In particular, for Q=B, we havewhich is the fundamental inequality for mixed quermassintegrals. For , and , the mixed Orlicz-quermassintegral has the following integral representation (see Ref.[14]):

For and , the mixed Orlicz-quermassintegral about three convex bodies is defined by(17)Here denotes the left derivative of ϕ(t) at t=1.

We will give the integral representation of in Section 2.

From (11), we see that if and , then the dual mixed volume is written as (the dual quermassintegral of K). In particular, and . The dual mixed quermassintegral has the following integral representation:(18)

Then, let us introduce the dual mixed quermassintegrals .

For and , we define the dual quermassintegrals by(19)

For , and , the dual mixed Orlicz-quermassintegrals has the following integral representation:

Let , and , we define the dual mixed Orlicz-quermassintegrals by(20)Here denotes the right derivative of at t=1.

2 The (Dual) Mixed Orlicz Quermassintegrals

Lemma 1   If and then for ,

Proof   We write as , and define by . Letand

Since , for , the existence of linf and lsup is obtained. By (15), thenand

The continuity of the mixed quermassintegral Wi implies that g is continuous at origin o. Thus(21)and(22)

The weak continuity of surface area measures as well as (see Ref. [13]) implies that

From (see Ref. [13]), we can obtain that(23)

Similarly, we have(24)Combining (21), (22), (23), and (24), we know that is differential at o. In fact, a bit more than will be proved, then . Therefore, is differential at o, furthermore,By (17), we can complete the proof of Lemma 1.

Remark 1   For Lemma 1, if , then by (2), and the case was introduced by Wang [9] in 2013. If Q=B, , then we have the integral representation of (see also Ref. [14]).

Theorem 4   If , and , then for ,If ϕ is strictly convex, the equality holds if and only if K and L are dilates.

Proof   If , then by (13) and (16), we have

Now, we verify the equality conditions. First, from the equality condition of Jensen’s inequality (13), the sufficiency is easy to prove, then we prove the necessity.

Suppose the equality holds. From the injectivity of ϕ, we have the equality in (16). Then, K and L are homothetic, so there exist and r>0 such that . Hence, by the definition of the support function, we haveAnd then we just have to prove that x=o. Since ϕ is strictly convex, by the equality condition of Jensen’s inequality, we havefor -almost all . Thus,Note that the centroid of is at the origin, so it follows thatThus x is the origin, and therefore K and L are dilates.

Remark 2   The case Q=B of Theorem 4 was established by Xiong and Zou [14], and when i=0, it is the Orlicz-Minkowski inequality (see Ref. [13]).

Lemma 2   Suppose , then for ,(25)In particular, .

Proof   By (7) and (8), we havethen using (18),Hence, by (19)

Lemma 3   [27] Let and . Thenuniformly for all .

Lemma 4   Let , and . Then(26)

Proof   Let , , and . From Lemma 3, we haveThen, by (18),Hence, by (20), we have

Theorem 5   Let , and . If is concave, thenwhile if is convex, the inequality is reversed. When is strictly concave (or convex, as appropriate), the equality holds if and only if K and L are dilates.

Proof   If is concave, by (26) and (13), it follows thatWhen is strictly concave, from the process of proving the equality of Theorem 4, we have that K and L are dilates.

Remark 3   For Theorem 5, taking Q=B and , , we can obtain an inequality which was established by Liu [22]. Furthermore, the case i=0 is the dual Orlicz-Minkowski inequality (see Refs. [17, 27]).

3 Proof of the Main Results

Theorem 6   Let , and . ThenIf ϕ is strictly convex, the equality holds if and only if K and L are dilates.

Proof   We write as . From (2), (14) and Lemma 1, it follows thatBy Theorem 4, we have

Thus, the proof of the inequality of this theorem is completed. From Theorem 4, the equality conditions can be obtained immediately.

Remark 4   For Theorem 6, the case Q=B and is (3).

Theorem 7   Let , and . If is concave, thenwhile if is convex, the inequality is reversed. When is strictly concave (or convex, as appropriate), the equality holds if and only if K and L are dilates.

Proof   We only prove the case in which is concave, and the case in which is convex is analogous. Let . If is concave, by (4), (18), (26) and Theorem 5, it follows thatWhen is strictly concave, from the process of proving the equality of Theorem 4, we know the equality holds if and only if that K and L are dilates.

Remark 5   For Theorem 7, taking Q=B and , , we can obtain an inequality which was established by Liu [22]. Furthermore, the case i=0 is the dual Orlicz-Minkowski inequality (see Refs. [17, 27]).

Lemma 5   [21] Let and . If , then

Proof of Theorem 1   Suppose . We clearly have that and, moreover, that is convex. From Theorem 7 (for K* and L*) together with Lemma 5, we getBy the equality condition of Theorem 7, equality holds if and only if K and L are dilates.

Remark 6   For Theorem 1, the case Q=B and is (5). Furthermore, when , , the case i=0 is stated by Firey [28].

Lemma 6   [22] Let and . If , then

Proof of Theorem 2   Without the loss of generality, we may consider that is concave. Suppose , , so . Thus, from Theorem 6 (for ϕ, and ) together with Lemma 7, we getwith equality if and only if K and L are dilates.

Remark 7   For Theorem 3, the case Q=B and is (6).

Lemma 7   [22] Let and such that is convex, then

Proof of Theorem 3   Suppose . We clearly have . From Theorem 6 (for and ) together with Lemma 6, we get

From the equality condition of Theorem 6, equality holds if and only if K and L are dilates.

Remark 8   For Theorem 2, the case Q=B and is (7).

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