Bodies with similar projections

Aleksandrov's projection theorem characterizes centrally symmetric convex bodies by the measures of their orthogonal projections on lower dimensional subspaces. A general result proved here concerning the mixed volumes of projections of a collection of convex bodies has the following corollary. If K is a convex body in R" whose projections on r-dimensional subspaces have the same r-dimensional volume as the projections of a centrally symmetric convex body A/, then the Quermassintegrals satisfy \Vj(M) Wj(K), for 0 < j < n -r, with equality, for any j, if and only if K is a translate of M. The case where K is centrally symmetric gives Aleksandrov's projection theorem.