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

Imre Barany; Liping Yuan

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arXiv · 1403.0437

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

Imre Barany·Liping Yuan

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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 $P \in \mathcal{P}^d(r)$, $d!\mathrm{vol\;} P$ takes all possible integer values between $cr^{d-1}$ and $r^d$ where $c>0$ is a constant depending only on $d$.

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Imre Barany, Liping Yuan. 2014-03-03. Volumes of convex lattice polytopes and a question of V. I. Arnold. https://arxiv.org/abs/1403.0437

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