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The longest segment in the complement of a packing

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

K. Böröczky Jr  and
G. Tardos
K. Böröczky Jr
Affiliation:
Rényi Institute of Mathematics, Budapest, P.O. Box 127, 1364 Hungary
G. Tardos
Affiliation:
Rényi Institute of Mathematics, Budapest, P.O. Box 127, 1364 Hungary
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Let K be a compact convex body in ℝn not contained in a hyperplane, and denote the norm whose unit ball is ½(K − K) by ║·║k. Given a translative packing of K, we are interested in how long a segment (with respect to ║·║K) can lie in the complement of the interiors of the translates. The main result of this note is to show the existence of a translative packing such that the length of the longest segments avoiding it is only exponential in the dimension n (see below). We start here with a lower bound, showing that this bound is close to optimal for balls.

MSC classification

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

Information

Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 45 - 49
Copyright
Copyright © University College London 2002

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References

1.Böröczky, K. and Soltan, V.. Translational and homothetic clouds for a convex body. Studia Sci. Math. Hung., 32 (1996), 93–102.Google Scholar
2.Henk, M. and Zong, Ch.. Segments in ball packings. Mathematika, 47 (2000), 31–38.CrossRefGoogle Scholar
3.Heppes, A.. Ein Satz über gitterförmige Kugelpackungen. Ann. Univ. Sci. Budapest Sect. Math., 3–4.(1960/1961), 89–90.Google Scholar
4.Kabatjanski, G. A. and Levenstein, V. I.. Bounds for packings on a sphere and in a space. Problems. Inform. Trans., 14 (1978), 1–17.Google Scholar
5.Rogers, C. A.. A note on coverings. Mathematika, 4 (1957), 1–6.CrossRefGoogle Scholar
6.Rogers, C. A. and Shephard, G. C.. The difference body of a convex body. Arch. Math., 8 (1957), 220–233.CrossRefGoogle Scholar
7.Schneider, R.. Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993).CrossRefGoogle Scholar
8.Talata, I.. On translational clouds for a convex body. Geometriae Dedicata, 80 (2000), 319–329.CrossRefGoogle Scholar
9.Zong, Ch.. A problem of blocking light rays. Geometriae Dedicata, 67 (1997), 117–128.CrossRefGoogle Scholar
10.Zong, Ch.. Sphere Packings. Springer, Berlin (1999).Google Scholar