Embedding a polytope in a lattice
The paper refers to the embeddability of the vertex sets of polytopes in lattices, i.e., to the theory of lattice polytopes. Using a certain similarity transform, the author proves e.g. the following results. If a \((4n-1)\)-dimensional polytope is similar to a lattice polytope in \(\mathbb{R}^{4n}\), then it is similar to a lattice polytope in \(\mathbb{R}^{4n- 1}\). Further on, an \(n\)-dimensional polytope \(P\) is similar to a lattice polytope in some \(\mathbb{R}^ N\) if and only if it is similar to a lattice polytope in \(\mathbb{R}^{2n+1}\), and if and only if \(\sin^ 2(<ABC)\) is rational for any three vertices \(A\), \(B\), \(C\) of \(P\).
scientific article; zbMATH DE number 764247