Dual Quermassintegral Differences for Intersection Body

In this paper, we introduce the concept of dual quermassintegral differences. Further, we give the dual Brunn-Minkowski inequality and dual Minkowski inequality for dual quermassintegral differences for mixed intersection bodies.


Introduction
The projection body was introduced in 1934 by Minkowski [1]. The research on the projection body has attracted much attention. The intersection operator and the class of intersection bodies were introduced in 1988 by Lutwak [2], who found a close connection between those bodies and famous Busemann-Petty problem (See [3]- [6]).
In [2], Lutwak presented the mysterious duality between projection and intersection bodies. For convex bodies K and L , let K Π and ( ) 1 , K L Π denote the projection body of K and mixed projection body of K and L , respectively. In [7], Lutwak established the following Brunn-Minkowski inequality for projection body and Minkowski inequality for mixed projection body: Theorem A. Let K and L be convex bodies in n  . Then with equality if and only if K and L are homothetic.
In [8], Theorem A and Theorem B were extended to volume differences: Theorem C. Suppose that K, L, and 1 D are convex bodies in n  , and 1 D K with equality if and only if K and L are homothetic and with equality if and only if K and L are homothetic.
For star bodies K and L , let IK and ( ) 1 , I K L denote the intersection body of K and mixed intersection body of K and L , respectively. In with equality if and only if L is a dilatate of K.
In this paper, we shall prove the dual forms of inequalities (1.3) and (1.4) for mixed intersection body. In this work new contributions that illustrate this mysterious duality will be presented. Our main results can be stated as follows: Theorem 1.1. Let , K L and 1 D are star bodies in n  and 1 2 , with equality if and only if L is a dilatate of K and 1 D are star bodies in n  and 1 2 , with equality if and only if L is a dilatate of K. Please see the next section for related definitions and notations.

Definitions and Notations
In this section, we will recall some basic results for dual quermassintegrals of star bodies. The reader is referred to Gardner [10], Lutwak [2] [11] and Thompson [12] for the Brunn-Minkowski theory with its dual theory. As usual, let n B denote the unit ball in Euclidean n -space, n  . While its boundary is 1 n S − and its volume is denoted by n k . For a compact subset K of n  , with o K ∈ , star-shaped with respect to o , the radial , K ρ ⋅ is continuous and positive, K will be called a star body. Two star bodies , The radial sum of two star bodies 1 2 , K K is defined as the star body K satisfying . This operation will be denoted by +  , i.e., 1 2 .
Let M and K be star bodies in n  . If K M ⊆ , then the dual i -quermassintegral difference function of 3), then we get the volume difference of star bodies M and K : (See [13] for the concept of the volume difference of two compact domains). The intersection body of a star body K, IK , is the centrally symmetric body whose radial function on 1 n S − is given by [2] ( ) ( ) where v is ( ) The mixed intersection body ( )

Inequalities for Dual Quermassintegral Differences
In this section, we will establish two inequalities for dual quermassintegral differences of star bodies, which are generalizations of Theorem 1.1 and Theorem 1.2 presented in introduction.
, n n n n n n n n n n n n n n n n n n n n n r r with equality if and only if L is a dilatate of K and ( ) ( ) ( ) ( ) Obviously, the case 0, in Theorem 3.3 is just Theorem 1.2. We will require some additional notations and two analytic inequalities to prove Theorem 3.1 and Theorem 3.3. Firstly, we define a function ( ) Φ is an indefinite form of its variables. with equality holds if and only if the coordinates of , x y are proportional. A proof of Lemma 3.4 can be found in [15]. The inequality (3.1) was first proved by Bellman [16] and is known as Bellman's inequality.
On the other hand, if 0, bc x d The proof is complete.