Volumes of Convex Lattice Polytopes and A Question of V. I. Arnold

Abstract

We show by a direct construction that there are at least exp${\{cV^{(d-1)/(d+1)}\}}$ convex lattice polytopes in ${\mathbb{R}^d}$ of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family ${\mathcal{P}^{d}(r)}$ (to be defined in the text) of convex lattice polytopes whose volumes are between 0 and r ^d/d!. Namely we prove that for ${P \in \mathcal{P}^{d}(r), d!}$ vol P takes all possible integer values between cr ^d–1 and r ^d where ${c > 0}$ is a constant depending only on d.