Generalized translation invariant valuations and the polytope algebra

TL;DR

This paper explores the space of generalized translation invariant valuations, extending convolution operations, and demonstrates that McMullen's polytope algebra is a subalgebra within this framework, linking geometric and algebraic structures.

Contribution

It introduces a partial convolution for generalized valuations and proves the embedding of the polytope algebra into this valuation space, extending previous algebraic structures.

Findings

McMullen's polytope algebra embeds into the space of generalized translation invariant valuations.

A partial convolution extends the smooth valuation convolution to a broader class.

For polytopes in general position, convolution matches the polytope algebra product.

Abstract

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.