Double-lattice packings of convex bodies in the plane
type: Item
instance of: scholarly article
title: Double-lattice packings of convex bodies in the plane
zbMATH Open document ID: 0709.52007
DOI: 10.1007/BF02187800
published in: Discrete \& Computational Geometry
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131127
review text: A convex body K in \(E^ 2\) is strictly convex if its boundary contains no line segment. A lattice packing of the plane is a family \(\{K_ i\}\) of convex bodies all congruent to a single convex body K in which each body is a translate of every other and the vectors of the translations form a lattice. A packing \({\mathcal P}\) is called a double-lattice packing if P is the union of two lattice packings \({\mathcal P}_ 1\) and \({\mathcal P}_ 2\) such that the 180\(\circ\) rotation about some point interchanges \({\mathcal P}_ 1\) and \({\mathcal P}_ 2\). A parallelogram inscribed in K is extensive if the length of each of its sides is at least half the length of K in the direction of the side. The authors prove that if K is strictly convex and if \({\mathcal P}\) is a maximum density double-lattice packing with copies of K, then \({\mathcal P}\) is generated by a minimum area extensive parallelogram inscribed in K. They also prove that every convex plane body K admits a double-lattice packing of the plane with density of at least \(\sqrt{3}/2\), extending a result of Mahler and Féjes Tòth.
Mathematics Subject Classification ID: 52C15
Mathematics Subject Classification ID: 11H31
zbMATH DE Number: 4165900
zbMATH Keywords: convex plane body
zbMATH Keywords: double-lattice packing of the plane
author: Greg Kuperberg
author: Włodzimierz Kuperberg
MaRDI profile type: Publication
cites work: Q5793893
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cites work: The theorem of Minkowski-Hlawka
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description: scientific article; zbMATH DE number 4165900