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

Autor:	Barany, Imre, Yuan, Liping
Rok vydání:	2014
Předmět:	Mathematics - Combinatorics Mathematics - Metric Geometry Primary 52B20 secondary 11H06
Druh dokumentu:	Working Paper
Popis:	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 $d$. Comment: 11 pages, 2 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1403.0437 Zobrazit plný text záznamu View this record from Arxiv