A Characterization of Some Mixed Volumes via the Brunn–Minkowski Inequality

We consider a functional F on the space of convex bodies in the n-dimensional Euclidean space, whose value at a convex body K is given by the integral of the area measure of K with respect to a given continuous function f. We prove that F satisfies an inequality of Brunn–Minkowski type if and only if f is the support function of a convex body, i.e., 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.