On Minkowski's inequality and its application

In the paper, we first give an improvement of Minkowski integral inequality. As an application, we get new Brunn-Minkowski-type inequalities for dual mixed volumes.

On the other hand, by using Minkowski's inequality for s >1 and t >1, respectively, we obtain with equality if and only if f(x) and g(x) are proportional, and with equality if and only if f(x) and g(x) are proportional. From (1.5), (1.6) and (1.7), (1.3) easily follows. From the equality conditions of (1.5), (1.6) and (1.7), the case of equality stated in (i) follows.
(ii) We have (st)/(pt) <1. Similar to the above proof, we have with equality if and only if either On the other hand, in view of Minkowski's inequality for the cases of 0 < s <1 and 0 < t <1, with equality if and only if f(x) and g(x) are proportional, and with equality if and only if f(x) and g(x) are proportional.

An application
The setting for this paper is n-dimensional Euclidean space ℝ n (n >2). Associated with a compact subset K of ℝ n , which is star-shaped with respect to the origin, is its radial function r(K, ·): S n -1 ℝ, defined for u S n -1 , by If r(K, ·) is positive and continuous, K will be called a star body. Let S n denote the set of star bodies in ℝ n . Letδ denote the radial Hausdorff metric, that is defined as follows: if K, L ∈ S n , thenδ(K, L) = |ρ K − ρ L | ∞ (see e.g. [3]).
For K i ∈ S n , the dual mixed volumes were given by Lutwak (see [6]), as The well-known Brunn-Minkowski-type inequality for dual mixed volumes can be stated as follows [6]: Theorem 2.1 Let K, L ∈ S n , and i < n -1. Then, with equality if and only if K and L are dilates. The inequality is reversed for i > n -1 and i ≠ n.
In the following, we establish new Brunn-Minkowski-type inequalities for dual mixed volumes.
Theorem 2.2 Let K, L ∈ S n and i, j, k ℝ.
Proof We begin with the proof of (i). From (2.1), (2.5) and (1.3), we havẽ with equality if and only if as stated in (i).