Lattices compatible with regular polytopes

In a recent paper, Karpenkov has classified the lattice polytopes (that is, with vertices in the integer lattice Z d ) which are regular with respect to those affinities which preserve the lattice. An alternative approach is adopted in this paper. For each regular polytope P in euclidean space E d , those lattices Λ are classified which are compatible with P , in the sense that some translate of Λ contains the vertices of P , and this translate is preserved by the symmetries of P .