Rotation equivariant Minkowski valuations

TL;DR

This paper characterizes the projection body operator as the unique non-trivial, rotation-equivariant, translation-invariant valuation on convex bodies in Euclidean space, highlighting its special role in convex geometry.

Contribution

It proves that the projection body operator is the only non-trivial operator with translation invariance, rotation equivariance, and valuation properties in convex geometry.

Findings

 The projection body operator is uniquely characterized by its properties.

 It maps polytopes to polytopes, preserving certain geometric structures.

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.