, Volume 24, Issue 2, pp 1064-1091
Date: 10 Oct 2012

A Characterization of Some Mixed Volumes via the Brunn–Minkowski Inequality

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Abstract

We consider a functional \(\mathcal{F}\) on the space of convex bodies in ℝ n of the form $$ {\mathcal{F}}(K)=\int_{\mathbb{S}^{n-1}} f(u) \mathrm{S}_{n-1}(K,du), $$ where \(f\in C(\mathbb{S}^{n-1})\) is a given continuous function on the unit sphere of ℝ n , K is a convex body in ℝ n , n≥3, and S n−1(K,⋅) is the area measure of K. We prove that \(\mathcal{F}\) satisfies an inequality of Brunn–Minkowski type if and only if f is the support function of a convex body, i.e., \(\mathcal{F}\) is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree n−1 and satisfy a Brunn–Minkowski type inequality.

Communicated Steven G. Krantz.
E. Saorín Gómez was supported by Direcciónn General de Investigación MTM2011-25377 and by “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, 04540/GERM/06.