On the number of lattice free polytopes
Given a polytope \(P\subset\mathbb{R}^d\), let vert \(P\) denote the set of its vertices. The polytope \(P\) is a lattice polytope if vert \(P\subset \mathbb{Z}^d\) and is lattice free if \(P\cap\mathbb{Z}^d =\text{vert} P\). Two lattice polytopes, \(P\) and \(Q\), are considered equivalent if there is a lattice-preserving affine transformation carrying \(P\) to \(Q\). Equivalent polytopes have the same volume. Denote \(N(d,v)\) the number of equivalence classes of lattice polytopes in \(\mathbb{R}^d\) of volume at most \(v\), \(E(d,v)\) (resp. \(M(d,v))\) the number of equivalence classes of lattice free simplices (resp. polytopes) in \(\mathbb{R}^d\) of volume at most \(v\). In an earlier paper the first author gave an estimate for \(N(d,v)\): \[ c_1v^{d-1 \over d+1}\leq\log N(d,v)\leq c_2v^{d-1\over d+1} \] where \(c_1\) and \(c_2\) are constants depending on \(d\). In this paper estimates for \(E(d,v)\) and \(M(d,v)\) are given: \[ \begin{aligned} c_1v^{d-1} & \leq E(d,v) \leq c_2v^{d-1}\\ M(d,v) & \leq cv^{2^d-1}(\log v)^{d-2}. \end{aligned} \] The proof of these results is based on the flatness theorem which gives an upper bound on the lattice width of a convex body.
scientific article; zbMATH DE number 1406281
scientific article