Embedding a polytope in a lattice

We present a special similarity ofR4n which maps lattice points into lattice points. Applying this similarity, we prove that if a (4n−1)-polytope is similar to a lattice polytope (a polytope whose vertices are all lattice points) inR4n, then it is similar to a lattice polytope inR4n−1, generalizing a result of Schoenberg [4]. We also prove that ann-polytope is similar to a lattice polytope in someRN if and only if it is similar to a lattice polytope inR2n+1, and if and only if sin2(<ABC) is rational for any three verticesA, B, C of the polytope.