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

Abstract

We show by a direct construction that there are at least $\exp \{c{V}^{(d-1)/(d+1)}\}$ 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 $P\in {\mathcal{P}}^d(r)$, $d!\mathrm{vol}P$ takes all possible integer values between $c{r}^{d-1}$ and $r^d$ where $c>0$ is a constant depending only on $d$.

11 pages, 2 figures