Abstract
We present a special similarity ofR 4n 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) inR 4n, then it is similar to a lattice polytope inR 4n−1, generalizing a result of Schoenberg [4]. We also prove that ann-polytope is similar to a lattice polytope in someR N if and only if it is similar to a lattice polytope inR 2n+1, and if and only if sin2(<ABC) is rational for any three verticesA, B, C of the polytope.
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Maehara, H. Embedding a polytope in a lattice. Discrete Comput Geom 13, 585–592 (1995). https://doi.org/10.1007/BF02574065
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DOI: https://doi.org/10.1007/BF02574065