Hadwiger-Wills-type higher-dimensional generalizations of Pick's theorem
type: Item
instance of: scholarly article
title: Hadwiger-Wills-type higher-dimensional generalizations of Pick's theorem
zbMATH Open document ID: 0960.52012
published in: Discrete \& Computational Geometry
publication date: 2001-01-17T00:00:00Z
review text: Pick's classical result gives a simple closed formula for the area of a (not necessarily convex) lattice polygon in terms of the interior lattice points and the lattice points on the boundary. This has been extended to non-proper lattice polygons by \textit{H. Hadwiger} and \textit{J. M. Wills} [J. Reine Angew. Math. 280, 61-69 (1976; Zbl 0321.52004)]. A polyhedron is a triangulated point set in~\(\mathbb R^d\). It is called non-proper if the underlying simplicial complex is not pure, that is, it has facets of varying dimension. Extensions of Pick's theorem to proper polyhedra of arbitrary dimension have been obtained by \textit{I. Macdonald} [Proc. Camb. Philos. Soc. 59, 719-726 (1963; Zbl 0126.18103)] and others. The author of the paper under review proves formulae for the volume of non-proper lattice polyhedra in dimensions three and four.
reviewed by: Michael Joswig
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1507681
zbMATH Keywords: lattice polyhedra
zbMATH Keywords: Pick's theorem
author: Krzysztof Kołodziejczyk
MaRDI profile type: Publication
full work available at URL: s004540010041
OpenAlex ID: W1999092613
DOI: 10.1007/S004540010041
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description: scientific article; zbMATH DE number 1507681