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Inequalities for dual affine quermassintegrals. (English) Zbl 1094.52004
Let $$K,L$$ be two star bodies in the $$n$$-dimensdional euclidean space containig the origin in their interiors; let $$\xi \in G(n,i)$$ and $$0\leq p \leq i$$; let $$\tilde V(K\cap \xi,i-p;L\cap \xi,p)$$ denote the dual mixed volume where $$K$$ appears $$i-p$$ times and $$L$$ appears $$p$$ times. The authors introduce the notion of mixed $$p$$-dual affine quermassintegrals in the following way: $\tilde {\Phi}_{p,i}(K,L)=k_n \left( \int_{G(n,n-i)} \left[ \frac{\tilde V_{p,n-i}(K,L;\xi)}{k_{n-i}} \right]^n d\xi \right)^{\frac 1n}$ where $\tilde V_{p,i}(K,L;\xi)=\tilde V(K\cap \xi,i-p;L\cap \xi,p).$ They establish the Minkowski inequality for them: $\tilde {\Phi}_{p,i}(K,L)^{n-i}\leq \tilde {\Phi}_{p,i}(K,K)^{n-i-p} \tilde {\Phi}_{n-i,i}(K,L)^p,$ with equality if and only if $$K$$ is a dilatate of $$L$$. As an application, the dual Brunn-Minkowski inequality for the dual affine quermassintegrals is obtained. They also establish a connection between the affine quermassintegrals and the dual affine quermassintegrals for a given convex body.

##### MSC:
 52A39 Mixed volumes and related topics in convex geometry 52A30 Variants of convex sets (star-shaped, ($$m, n$$)-convex, etc.) 52A40 Inequalities and extremum problems involving convexity in convex geometry
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