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

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

2012-02-10

arxiv.org/abs/1202.2241v1

Mathematics

Metric Geometry

24 pages

Scientific paper

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.

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