On the maximal width of empty lattice simplices
type: Item
instance of: scholarly article
title: On the maximal width of empty lattice simplices
zbMATH Open document ID: 0966.52013
author: Christian Haase
author: Günter M. Ziegler
published in: European Journal of Combinatorics
publication date: 2001-08-16T00:00:00Z
review text: A \(k\)-dimensional lattice simplex \(S\subset\mathbb{R}^d\) is the convex hull of \(k+1\) affinely independent points from \(\mathbb{Z}^d\), and \(S\) is said to be empty if it intersects \(\mathbb{Z}^d\) only at its vertices. Unexpectedly, the authors construct infinitely many empty lattice \(d\)-simplices of arbitrarily large volume from \((d-1)\)-dimensional ones of the same lattice width, yielding in particular also an infinite sequence of empty lattice 4-simplices of width 2. Modifying a former conjecture they formulate the following finiteness conjecture: For every \(d\geq 2\), there are only finitely many equivalence classes of empty \(d\)-simplices whose width is larger than the greatest width that can be achieved by almost empty simplices in dimension \(d-1\). (A lattice simplex without interior lattice points and with at least one empty facet is called almost empty).
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1406282
zbMATH Keywords: empty lattice simplex
zbMATH Keywords: width function
zbMATH Keywords: lattice polytope
zbMATH Keywords: unimodular equivalence
zbMATH Keywords: lattice simplex
MaRDI profile type: Publication
OpenAlex ID: W2045230236
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DOI: 10.1006/EUJC.1999.0325
description: scientific article; zbMATH DE number 1406282