The integral polytope group (Q2664122)

The set of polytopes in a real finite-dimensional space \(V\) forms a commutative monoid under the Minkowski sum. Let \(\mathcal{P}(V)\) denote its Grothendieck group. Given a finitely generated free-abelian group \(H\), a polytope in \(V_H=H\otimes_{\mathbb Z} \mathbb{R}\) is said to be integral if all of its vertices belong to \(H\). Then the subgroup \(\mathcal{P}(H) \subset \mathcal{P}(V_H)\) is determined and called the integral poytope group. The author shows that the Grothendieck group \(\mathcal{P}(\mathbb{Z}^n)\) associated to integral polytopes in \(\mathbb{R}^n\) is free abelian, by providing an explicit basis. Moreover, the author identifies the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. The author also computes the kernel of the norm map sending a polytope to its induced seminorm on the dual of \(\mathbb{R}^n\).