On the number of lattice free polytopes
type: Item
instance of: scholarly article
title: On the number of lattice free polytopes
zbMATH Open document ID: 0954.52015
author: Jean-Michel Kantor
author: Imre Bárány
published in: European Journal of Combinatorics
publication date: 2001-02-14T00:00:00Z
review text: Given a polytope \(P\subset\mathbb{R}^d\), let vert \(P\) denote the set of its vertices. The polytope \(P\) is a lattice polytope if vert \(P\subset \mathbb{Z}^d\) and is lattice free if \(P\cap\mathbb{Z}^d =\text{vert} P\). Two lattice polytopes, \(P\) and \(Q\), are considered equivalent if there is a lattice-preserving affine transformation carrying \(P\) to \(Q\). Equivalent polytopes have the same volume. Denote \(N(d,v)\) the number of equivalence classes of lattice polytopes in \(\mathbb{R}^d\) of volume at most \(v\), \(E(d,v)\) (resp. \(M(d,v))\) the number of equivalence classes of lattice free simplices (resp. polytopes) in \(\mathbb{R}^d\) of volume at most \(v\). In an earlier paper the first author gave an estimate for \(N(d,v)\): \[ c_1v^{d-1 \over d+1}\leq\log N(d,v)\leq c_2v^{d-1\over d+1} \] where \(c_1\) and \(c_2\) are constants depending on \(d\). In this paper estimates for \(E(d,v)\) and \(M(d,v)\) are given: \[ \begin{aligned} c_1v^{d-1} & \leq E(d,v) \leq c_2v^{d-1}\\ M(d,v) & \leq cv^{2^d-1}(\log v)^{d-2}. \end{aligned} \] The proof of these results is based on the flatness theorem which gives an upper bound on the lattice width of a convex body.
reviewed by: Charles Leytem
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1406281
zbMATH Keywords: lattice free polytope
zbMATH Keywords: lattice width
zbMATH Keywords: volume
zbMATH Keywords: flatness
zbMATH Keywords: lattice polytope
MaRDI profile type: Publication
full work available at URL: eujc.1999.0324
OpenAlex ID: W2004414619
cites work: Q3871493
cites work: On the number of convex lattice polytopes
cites work: Q5340226
cites work: Convexity in cristallographical lattices
cites work: Q5512461
cites work: Geometric algorithms and combinatorial optimization
cites work: Q3259107
cites work: Triangulations of integral polytopes and Ehrhart polynomials
cites work: Q5785796
cites work: Covering minima and lattice-point-free convex bodies
cites work: A bound, in terms of its volume, for the number of vertices of a convex polyhedron when the vertices have integer coordinates
cites work: Q3745276
cites work: The difference body of a convex body
cites work: Integral Polyhedra in Three Space
DOI: 10.1006/EUJC.1999.0324
description: scientific article; zbMATH DE number 1406281