Rigid motion invariant valuations on polytopes

Abstract

We show that any measurable, translation and $\mathrm{SO}(n)$-invariant valuation on polytopes in ${\mathbb{R}}^n$ is a linear combination of the intrinsic volumes, which extends Hadwiger's classical characterization of rigid motion invariant continuous valuations on convex bodies. This result is based on a novel regularity result for translation invariant, measurable, $1$-homogeneous, and simple valuations. As a consequence, we obtain a similar characterization of the Steiner point map on polytopes.

arXiv comment29 pages