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

TL;DR

This paper characterizes certain mixed volumes via the Brunn-Minkowski inequality, showing that functionals satisfying this inequality correspond to support functions of convex bodies, thus linking geometric measures to convex analysis.

Contribution

It provides a new characterization of mixed volumes through a Brunn-Minkowski type inequality, connecting functional inequalities to convex geometric structures.

Findings

Functional $\\mathcal F$ satisfying Brunn-Minkowski inequality implies $f$ is a support function.

Characterization of translation invariant, continuous valuations homogeneous of degree $n-1$.

Establishes a link between inequalities and convex geometric measures.

Abstract

We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$ 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.