Bodies with similar projections

Abstract

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 K is a convex body in R n {\mathbb {R}}^{n} whose projections on r r -dimensional subspaces have the same r r -dimensional volume as the projections of a centrally symmetric convex body M M , then the Quermassintegrals satisfy W j ( M ) ≥ W j ( K ) W_{j}(M)\ge W_{j}(K) , for 0 ≤ j > n − r 0\le j > n-r , with equality, for any j j , if and only if K K is a translate of M M . The case where K K is centrally symmetric gives Aleksandrov’s projection theorem.