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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
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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References

1.Böröczky, K.. Über Dunkelwolken. Proc. Coll. Convexity (ed. Fenchel, W.), Copenhagen Univ. (1967), 13–17.Google Scholar
2.Böröczky, K. and Soltan, V.. Translational and homothetic clouds for a convex body. Studia Sci. Math. Hungar., 32 (1996), 93–102.Google Scholar
3.Csóka, G.. The number of congruent spheres that hide a given sphere of three-dimensional space is not less than 30. Studia Sci. Math. Hungar., 12 (1977), 323–334.Google Scholar
4.Danzer, L.. Drei Beispiele zu Lagerungsproblemen. Arch. Math., 11 (1960), 159–165.CrossRefGoogle Scholar
5.Tóth, L. Fejes. Verdeckung einer Kugel durch Kugeln. Publ. Math. Debrecen, 6 (1959), 234–240.CrossRefGoogle Scholar
6.Heppes, A.. Ein Satz über gitterförmiger Kugelpackungen. Ann. Univ. Sci. Budapest Eötvös Sect. Math., 3 (1961), 89–90.Google Scholar
7.Heppes, A.. On the number of spheres which can hide a given spheres. Canad. J. Math., 19 (1967), 413–418.CrossRefGoogle Scholar
8.Hortobágyi, I.. Durchleuchtung gitterförmiger Kugelpackungen mit Lichtbündeln. Studia Sci. Math. Hungar., 6 (1971), 147–150.Google Scholar
9.Horváth, J.. Über die Durchsichtigkeit gitterförmiger Kugelpackungen. Studia Sci. Math. Hungar., 5 (1970), 421–426.Google Scholar
10.Horvath, J. and Ryškov, S. S.. Estimation of the radius of a cylinder that can be imbedded in any lattice packing of n-dimensional unit balls. Mat. Zamelki, 17 (1975), 123–128.Google Scholar
11.Martini, H. and Soltan, V.. Combinatorial problems on the illumination of convex bodies. Aequationes Math., 57 (1999), 121–152.CrossRefGoogle Scholar
12.Rogers, C. A.. Packing and Covering (Cambridge University Press, Cambridge, 1964).Google Scholar
13.Talata, I.. On translational clouds for a convex body. Geom. Dedicata, 80 (2000), 319–329.CrossRefGoogle Scholar
14.Zong, C.. A problem of blocking light rays. Geom. Dedicata, 67 (1997), 117–128.CrossRefGoogle Scholar
15.Zong, C.. A note on Hornrich's problem. Arch. Math., 72 (1999), 127–131.CrossRefGoogle Scholar
16.Zong, C.. Sphere Packings (Springer-Verlag, New York, 1999).Google Scholar