On the connection between the projection and the extension of a parallelotope (Q883119)

With a parallelotope \(P\) in \({\mathbb R}^n\) (a polytope tiling \({\mathbb R}^n\) by translations) and a direction \(v\), the author associates a certain sublattice \(L_{v}\) of the packing lattice \(L\) of \(P\), depending on the face lattice of \(P\) and the direction \(v\). Let \(H\) be a hyperplane transversal to \(v\), and let \(S(v)\) be a nondegenerate segment parallel to \(v\). It is proved that the polytope \(P+S(v)\) is a parallelotope if and only if the projection of \(P\) to \(H\) along the direction \(v\) is a parallelotope in \(H\) with respect to the projection of the lattice \(L_{v}\).