Concavity of powers of a convolution

Then F, G and F x G are convex sets. Denote (x1* . . X, wl Wpyl * * Yn, VI ... vq) by (.x, w-,y, v-) and for z E Rn let H. be the affine space {(x, w, y-, v): x+ y= z). Let J. be the affine space {(x-, wy, v): x = a}. For arbitrary z and a, F x G n Hn Ja is either empty or the Cartesian product of a p-dimensional open cube of volume f(a) with a q-dimensional open cube of volume g(z a. In either case the (p + q)dimensional volume of Hz n J4, nF x G is f(a)g(z--). Since Hz is the disjoint union of the Hi n J4 and the distance between flats H. n Ja and Hi n Ja is VX ii a'ij, the (p + q + n)-dimensional volume V(z) of F x G n Hz is given by