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Segments in ball packings

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

M. Henk  and
C. Zong
M. Henk
Affiliation:
Abteilung für Analysis, TU Wien, Wiedner Hauptstr. 8–10/1142, 1040 Wien, Austria. E-mail: henk@tuwien.ac.at
C. Zong
Affiliation:
Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, P.R. China. E-mail: cmzong@math08.math.ac.cn
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Abstract

Denote by Bn the n-dimensional unit ball centred at o. It is known that in every lattice packing of Bn there is a cylindrical hole of infinite length whenever n≥3. As a counterpart, this note mainly proves the following result: for any fixed ε with ε>0, there exist a periodic point set P(n, ε) and a constant c(n, ε) such that Bn + P(n, ε) is a packing in Rn, and the length of the longest segment contained in Rn\{int(εBn) + P(n, ε)} is bounded by c(n, ε) from above. Generalizations and applications are presented.

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions

Information

Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 31 - 38
Copyright
Copyright © University College London 2000

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