Rotation equivariant Minkowski valuations

Abstract

The projection body operator Π, which associates with every convex body in Euclidean space n its projection body, is a continuous valuation, it is invariant under translations, and equivariant under rotations. It is also well known that Π maps the set of polytopes in n into itself. We show thatΠ is the only nontrivial operator with these properties.