Lattices compatible with regular polytopes

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