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Thin non-lattice covering with an affine image of a strictly convex body

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

Gábor Fejes Tóth  and
Wlodzimierz Kuperberg
Gábor Fejes Tóth
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary.
Wlodzimierz Kuperberg
Affiliation:
Department of Mathematics, Auburn University, AL 36849-5310, U.S.A.
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Abstract

We prove that for every strictly convex body C in the Euclidean space of dimension d≥3, some aflfine image of C admits a non-lattice covering of the space, thinner than any lattice covering. We illustrate the general construction with an example of a thin non-lattice covering of with certain congruent ellipsoids.

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 239 - 250
Copyright
Copyright © University College London 1995

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References

1.Bambah, R. P.. On lattice coverings by spheres. Proc. Nat. Inst. Sci. India, 20 (1954), 25–52.Google Scholar
2.Barnes, E. S.. The covering of space by spheres. Canad. J. Math., 8 (1956), 293–304.CrossRefGoogle Scholar
3.Bezdek, A. and Kuperberg, W.. Packing Euclidean space with congruent cylinders and with congruent ellipsoids. In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, edited by Gritzmann, P. and Sturmfels, B. (1991), 71–80.CrossRefGoogle Scholar
4.Coxeter, H. S. M., Few, L. and Rogers, C. A.. Covering space with equal spheres. Mathematika, 6 (1959), 147–157.CrossRefGoogle Scholar
5.Tóth, G. Fejes, Gritzmann, P. and Wills, J. M.. Sausage-skin problems for finite coverings. Mathematika, 31 (1984), 117–136.CrossRefGoogle Scholar
6.Tóth, G. Fejes and Kuperberg, W.. Packing and covering with convex sets. In Handbook of Convex Geometry, edited by Gruber, P. M. and Wills, J. M. (North-Holland, 1993), pp. 799–860.CrossRefGoogle Scholar
7.Tóth, L. Fejes. Some packing and covering theorems. Ada Sci. Math. Szeged, 12/A (1950), 62–67.Google Scholar
8.Tóth, L. Fejes. Regular figures (Pergamon Press, New York, 1964).Google Scholar
9.Few, L.. Covering space by spheres. J. London Math. Soc, 39 (1964), 51–54.CrossRefGoogle Scholar
10.Gauss, C. F.. Untersuchungen über die Eigenschaften der positiven ternàren quadratischen Formen von Ludwig August Seber. In Göttingische gelehrte Anzeigen, Juli 9, 1831=J. Reine angew. Math., 20 (1840), 312-320 = Werke II (1876), 188–196.Google Scholar
11.Rogers, C. A.. The closest packing of convex two-dimensional domains. Ada Math., 86 (1951), 309–321.Google Scholar
12.Wills, J. M.. An ellipsoid packing in E3 of unexpected high density. Mathematika, 38 (1991), 318–320.CrossRefGoogle Scholar