EMBEDDING A POLYTOPE IN A LATTICE

We present a special similarity of R(4n) which maps lattice points int o 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) in R(4n) then it is similar to a lattice poly tope in R(4n-1), generalizing a result of Schoenberg [4]. We also prov e that an n-polytope is similar to a lattice polytope in some R(N) if and only if it is similar to a lattice polytope in R(2n+1), and if and only if sin(2)(angle ABC) is rational for any three vertices A, B, C of the polytope.