Volumes of convex lattice polytopes and a question of V. I. Arnold (Q485021)

The authors show that the number of convex lattice polytopes in \(\mathbb{R}^d\) of volume~\(V\) is at least \(\exp(cV^{(d-1)/(d+1)})\), with \(c\) a positive constant, up to equivalence under lattice-preserving affine transformations. The estimate is derived from a statement about certain families of convex lattice polytopes \(\mathcal{P}^{d}(r)\), \(r \in \mathbb{N}\), saying that there exists a positive constant \(C\) such that all integers in the interval \([Cr^{d-1},r^d]\) occur as volumes of polytopes in \(\mathcal{P}^{d}(r)\).