The integral polytope group

TL;DR

This paper proves that the group of integral polytopes in Euclidean space is free-abelian, identifies an involution as a sum of Euler characteristic types, and computes the kernel of a related norm map.

Contribution

It provides an explicit basis for the integral polytope group, characterizes an involution as an Euler characteristic sum, and calculates the kernel of the polytope norm map.

Findings

The integral polytope group is free-abelian.

The involution corresponds to a sum of Euler characteristic types.

The kernel of the norm map is explicitly computed.

Abstract

We show that the Grothendieck group associated to integral polytopes in $\mathbb{R}^n$ is free-abelian by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of $\mathbb{R}^n$.