The integral polytope group

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 ${\mathbb{R}}^n$.