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 
Mathematics
CLC number: O193
The Mixed Polar OrliczBrunnMinkowski Inequalities
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan
411201, Hunan, China
^{†} To whom correspondence should be addressed. Email: wwang@hnust.edu.cn
Received:
5
December
2021
Some OrliczBrunnMinkowski type inequalities for (dual) quermassintegrals of polar bodies and star dual bodies have been introduced. In this paper, we generalize the results and establish some OrliczBrunnMinkowski type inequalities for mixed (dual) quermassintegrals of polar bodies and star dual bodies.
Key words: polar body / mixed quermassintegral / OrliczBrunn Minkowski inequalities
Biography: LI Juan, female, Master candidate, research direction: convex geometry. Email: lj2745675651@163.com
Foundation item: Supported by the Natural Science Foundation of Hunan Province (2021JJ30235)
© Wuhan University 2022
This 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 ndimensional Euclidean space R ^{ n }. For , their Minkowski sum , and the ith quermassintegral of K will be denoted by for each . The classical BrunnMinkowski 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 BrunnMinkowski inequality (see Ref. [1]). It is the core of the BrunnMinkowski 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 L_{p} addition. Let be the set of all convex bodies in containing the origin in their interiors. For R ^{ n }, and , the L_{p} Minkowski addition +_{p} is defined by (see Ref. [2])where h_{K} 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 L_{p} addition, Lutwak^{[3,4]} established the L_{p} BrunnMinkowski inequality for quermassintegrals: For , and n1, thenwith equality if and only if K and L are dilates. Readers can refer to Refs. [59] for additional references.
The OrliczBrunnMinkowski theory originated from the work of Lutwak et al in 2010^{[10,11]}. As an important part of the theory, the Orlicz BrunnMinkowski inequality has been very popular with scholars in related fields. At first, the Orlicz BusemannPetty 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 BrunnMinkowski inequality that implied the L_{p} BrunnMinkowski 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 OrliczBrunnMinkowski 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 BrunnMinkowski theory for star bodies. In 2015, Gardner et al ^{[17]} established the dual OrliczBrunnMinkowski 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 BrunnMinkowskitype in equality for the polar set of the psum of convex bodies, which generalized previous results by Firey^{[20]}; Wang and Huang ^{[21]} gave a systematic explanation of Orlicz BrunnMinkowski inequality for polar bodies and dual star bodies and Liu ^{[22]} established some OrliczBrunn Minkowski type inequalities for (dual) quermassintegrals of polar bodies and star dual bodies. Besides, the OrliczBrunnMinkowski 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 OrliczBrunnMinkowski 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 OrliczBrunnMinkowski type inequality for dual mixed quermassintegrals of polar bodies and star dual bodies.
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 BrunnMinkowski 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 OrliczBrunn 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 AleksandrovFenchel inequality (with respect to three convex bodies) can be generalized to Orlicz setting. In Section 3, we introduce some OrliczBrunnMinkowski 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 L_{p} 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 starshaped set K about the origin, the radial function is defined bywhere the compact starshaped 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 starshaped 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 ^{ n1}.
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 μintegrable 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 μalmost all (see Refs. [22, 26]). If ϕ is a concave function, the inequality is reversed.
For a convex body K, the ith 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 AleksandrovFenchel 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 Orliczquermassintegral has the following integral representation (see Ref.[14]):
For and , the mixed Orliczquermassintegral 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 Orliczquermassintegrals has the following integral representation:
Let , and , we define the dual mixed Orliczquermassintegrals by(20)Here denotes the right derivative of at t=1.
2 The (Dual) Mixed Orlicz Quermassintegrals
Proof We write as , and define by . Letand
Since , for , the existence of l_{inf} and l_{sup} is obtained. By (15), thenand
The continuity of the mixed quermassintegral W_{i} 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 OrliczMinkowski 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 .
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 OrliczMinkowski 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 OrliczMinkowski 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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