  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
$\mathbb{R}^n$.
