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arXiv · 2608.19110

Rigid motion invariant valuations on polytopes

Jonas Knoerr
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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.

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Jonas Knoerr. 2026-08-19. Rigid motion invariant valuations on polytopes. https://arxiv.org/abs/2608.19110

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