The Mixed Polar Orlicz-Brunn-Minkowski Inequalities

: Some Orlicz-Brunn-Minkowski 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 Orlicz-Brunn-Minkowski type inequalities for mixed (dual) quermassintegrals of polar bodies and star dual bodies.


Introduction
Let n K be the set of convex bodies (compact convex sets with nonempty interior) in the n -dimensional Euclidean space n R .For , , and the i -th quermassintegral of K will be denoted by ( ) . The classical Brunn-Minkowski inequality for quermassintegrals states that for , with equality if and only if K and L are homothetic.The case 0 i  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 p L -addition.Let o n K be the set of all convex bodies in containing the origin in their interiors.For n R , o , n K L K  and 1 p  , the p L -Minkowski addition p  is defined by (see Ref. [2]) where K h denotes the support function of the convex body K and it is defined by ( ) sup{ : }.

K h x
x y y K    Here x, y denote the standard inner product of x, n y  R .Thirty years after the new p L -addition, Lutwak [3,4] established the p L -Brunn-Minkowski inequality for quermassintegrals: with equality if and only if K and L are dilates.
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 p L -Brunn-Minkowski inequality in 2014.Let  be the class of convex and strictly increasing functions, :[0, ) [0, ) for n x  R .In the same year, Xiong and Zou [14] established the Orlicz-Brunn-Minkowski inequality for quermassintegrals: If  is strictly convex, equality holds if and only if K and L are dilates.The case 0 i  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, :[0, ) [0, ) for \ {0} n x  R .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 inequality 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 K  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 o , , If  is strictly convex, equality holds if and only if K and L are dilates.For o , , 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 o , , , , ,  is convex, the inequality is reversed.If 0  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 o , ,  is strictly convex, and 0 ) 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. , with 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.

Preliminaries
, and by ( ) 0 In particular, if Gardner, Hug and Weil [13] proved that Orlicz addition is commutative if and only if   .Therefore, (2) was defined.
For a compact star-shaped set K about the origin, the radial function : where the compact star-shaped set n K  R is defined if the intersection of every straight line through the origin with K is a line segment.And if the K  is positive and continuous, then the compact star-shaped set K about the origin is called a star body. For Equivalently, the radial Orlicz addition   % can be defined implicitly by Then by the corresponding special case Therefore, (4) was defined.
We denote the unit ball in n R and its surface by 1 , n B S  , respectively.The dual mixed volume 1 ( , , ) where S is the spherical Lebesgue measure ((n−1) dimensional Hausdorff measure) of and it is easy to see that K  is a convex body and ( )  (a convex body that contains the origin in its interior), for all Suppose that  is a probability measure on a space X and : where I is a possible infinite interval.Jensen's inequality states that if When  is strictly convex, equality holds if and only if ( ) g x is a constant for  -almost all x X  (see Refs. [22, 26]).If  is a concave function, the inequality is reversed.For a convex body K, the i -th quermassintegral of K, ( ) , where B is the unit ball in n R , and V , S denote the volume and the surface area of the set involved, respectively.


denotes the left derivative of ( ) t We will give the integral representation of , ( , , ) From (11), we see that if 1 , then the dual mixed volume ( , , , , , ) . The dual mixed quermassintegral Then, let us introduce the dual mixed quermassintegrals ( , , ) we define the dual mixed Orlicz-quermassintegrals , ( , , ) Here ( 1) denotes the right derivative of ( ) t

The (Dual) Mixed Orlicz Quermassintegrals
existence of inf l and sup l is obtained.By (15), then The continuity of the mixed quermassintegral i W implies that g is continuous at origin o .Thus and The weak continuity of surface area measures as well as (see Ref. [13]) implies that Ref. [13]), we can obtain that Similarly, we have ( )d ( , , ) ( ) Combining ( 21), ( 22), (23), and ( 24), we know that ( ) g  is differential at o .In fact, a bit more than inf sup l l  will be proved, then inf sup l l  .Therefore, By (17), we can complete the proof of Lemma 1. 2), and the case p  was introduced by Wang [9] in 2013.If Q B  , 0 1 i n    , then we have the integral representation of ( , ) Ref. [14]).
, , , , , If  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  , we have the equality in (16).Then, K and L are homothetic, so there exist  is at the origin, so it follows that Thus 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 0 i  , it is the Orlicz-Minkowski inequality (see Ref. [13]).
Lemma 2 Suppose o , , ) In particular, Proof By ( 7) and ( 8), we have Proof Let 0 From Lemma 3, we have Then, by (18), , , , , ,  is strictly concave (or convex, as appropriate), the equality holds if and only if K and L are dilates.
and ( 13), it follows that ,  is strictly concave, from the process of prov- ing 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 0 i  is the dual Orlicz-Minkowski inequality (see Refs. [17, 27]).

Theorem 6
Let o , , By 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 0 concave, by ( 4), ( 18), (26) and Theorem 5, it follows that ,  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 0 i  is the dual Orlicz-Minkowski inequality (see Refs. [17, 27]).
Lemma 5 [21] Let We clearly have that     and, moreover, that 1 1 0 ( ) and L  ) together with Lemma 5, we get , By 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 0 1 i n    is (5).Furthermore, when ( ) p t t   , 1 p  , the case 0 i  is stated by Firey [28] .Lemma 6 [22] with equality if and only if K and L are dilates.Remark 7 For Theorem 3, the case Q B  and 0 1 i n    is (6).
Lemma 7 [22] , 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 0 1 i n    is (7).
Orlicz addition   (see Section 1 for precise definition) is defined by And then we just have to prove that x o  .Since  is strictly convex, by the equality condition of Jensen's inequality, we have

2 
Without the loss of generality, we may consider that 0  .Thus, from Theorem 6 (for  , K  and L  ) together with Lemma 7, we get strictly convex, the equality holds if and only if K and L are dilates.
the inequality is reversed.