We consider a class of integral functionals defined in the family of convex bodies. The value of the functional on a convex body is given by the integral of a fixed continuous function defined on the unit sphere, with respect to the area measure of the convex body. We assume that a functional of this form verifies an inequality of Brunn-Minkowski type. We prove that if in addition the functional is symmetric, then it must be a mixed volume. The same result holds if the function defining the functional has some regularity property.