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

We consider a functional $\\mathcal F$ on the space of convex bodies in $\\R^n$\ndefined as follows: ${\\mathcal F}(K)$ is the integral over the unit sphere of a\nfixed continuous functions $f$ with respect to the area measure of the convex\nbody $K$. We prove that if $\\mathcal F$ satisfies an inequality of\nBrunn--Minkowski type, then $f$ is the support function of a convex body, i.e.,\n$\\mathcal F$ is a mixed volume. As a consequence, we obtain a characterization\nof translation invariant, continuous valuations which are homogeneous of degree\n$n-1$ and satisfy a Brunn--Minkowski type inequality.\n