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 rd/d!. Namely we prove that for \({P \in \mathcal{P}^{d}(r), d!}\) vol P takes all possible integer values between crd–1 and rd where \({c > 0}\) is a constant depending only on d.