  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.
