A characterization of some mixed volumes via the Brunn-Minkowski inequality

Abstract

We consider a functional $\mathcal{F}$ on the space of convex bodies in ${\mathbb{R}}^n$ defined as follows: $\mathcal{F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body $K$. We prove that if $\mathcal{F}$ satisfies an inequality of Brunn--Minkowski type, then $f$ 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.

arXiv comment24 pages

References in corpus (1)

• Functional inequalities derived from the Brunn-Minkowski inequalities for quermassintegrals