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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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