Rotation equivariant Minkowski valuations

Abstract

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