Expandability radius versus density of a lattice packing

A. Heppes

doi:10.1023/a:1022345929586

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Periodica Mathematica Hungarica

Volume/Issue:: Volume 45: Issue 1-2

Expandability radius versus density of a lattice packing

Author:

A. Heppes

A. Heppes 1124 Budapest Vércse u. 24/A 1124 Budapest Vércse u. 24/A

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Pages:: 65–71

Online Publication Date:: 03 Jan 2005

Publication Date:: 01 Sep 2002

Article Category:: Research Article

DOI:: https://doi.org/10.1023/a:1022345929586

Keywords:: lattice packing; density; disc; efficiency

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Abstract

In the present paper lattice packings of open unit discs are considered in the Euclidean plane. Usually, efficiency of a packing is measured by its density, which in case of lattice packings is the quotient of the area of the discs and the area of the fundamental domain of the packing. In this paper another measure, the expandability radius is introduced and its relation to the density is studied. The expandability radius is the radius of the largest disc which can be used to substitute a disc of the packing without overlapping the rest of the packing. Lower and upper bounds are given for the density of a lattice packing of given expandability radius for any feasible value. The bounds are sharp and the extremal configurations are also presented. This packing problem is related to a covering problem studied by Bezdek and Kuperberg [BK97].

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Periodica Mathematica Hungarica

Print ISSN:: 0031-5303

Online ISSN:: 1588-2829

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Periodica Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1971
Volumes per Year	2
Issues per Year	4
Founder	Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's Address	H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0031-5303 (Print)
ISSN	1588-2829 (Online)