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Double packing of spheres: a new upper bound

Published online by Cambridge University Press:  26 February 2010

L. Few
L. Few
Affiliation:
University College, London
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Let S be a set of points in n-dimensional space, and suppose that an open sphere of unit radius is centred at each point of S. Suppose that no point of space is an inner point of more than two spheres. We say that S provides a double packing for spheres of unit radius. We define δ2(S), the density of this double packing, to be

where Jn is the volume of a sphere of unit radius, and Nt(S) is the number of points of S inside a cube of side 2t, centred at the origin O. We define δ2, the density of closest double packing, to be

where the supremum is taken over all sets S with the property described above.

Type
Research Article
Information
Mathematika , Volume 15 , Issue 1 , June 1968 , pp. 88 - 92
Copyright
Copyright © University College London 1968

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References

1.Blichfeldt, H. F., Math. Ann., 101 (1929), 605–608.CrossRefGoogle Scholar
2.Few, L., J. Lond. Math. Soc., 28 (1953), 297–304.CrossRefGoogle Scholar
3.Few, L., J. Lond. Math. Soc., 39 (1964), 51–54.CrossRefGoogle Scholar
4.Few, L., Proc. Coll. Convexity, Copenhagen, 1965 (1967), 88–93.Google Scholar
5.Rogers, C. A., Proc. Lond. Math. Soc. (3), 8 (1959), 609–620.Google Scholar