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A new bound on the local density of sphere packings. (English) Zbl 0787.52010

Continuing a number of results on and improvements of upper bounds on the density of (nonoverlapping) packings of unit spheres in 3-space, the author shows that such a density is at most \(0,773055\dots\); for showing this, he uses the Voronoi polyhedra of such packings. (Such a Voronoi polyhedron \(P\) is the set of points that lie closer to the midpoint of the corresponding sphere \(S\) than to the center of another sphere from the packing. Thus, the volume ratio \(V(S)\over V(P)\) is a local measure of density, and any upper bound on the local densities is an upper bound on the density of the packing as a whole.)
Namely, the author shows that a Voronoi polyhedron defined by such a sphere packing must have volume at least \(5,41848\dots\); by cutting the Voronoi polyhedron into cones (one for each of its facets), a lower bound on the cone volumes is established as a function of the respective solid angle. The sum of all the cone volume bounds is minimized when there are 13 facets, each of solid angle \(4\pi\over 13\).

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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References:

[1] J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices, and Groups, Springer-Verlag, New York, 1988. · Zbl 0634.52002 · doi:10.1007/978-1-4757-2016-7
[2] Wu-Yi Hsiang, On the sphere packing problem and the proof of Kepler’s conjecture, Preprint, 1992. · Zbl 0887.52011
[3] J. H. Lindsey II, Sphere packing inR3,Mathematika33 (1986), 137-147. · Zbl 0582.52007 · doi:10.1112/S0025579300013954
[4] J. H. Lindsey II, Sphere packing, Preprint, 1987.
[5] D. J. Muder, Putting the best face on a Voronoi Polyhedron,Proc. London Math. Soc. (3)56 (1988), 329-348. · Zbl 0609.52012 · doi:10.1112/plms/s3-56.2.329
[6] C. A. Rogers,Packing and Covering, Cambridge University Press, Cambridge, 1964. · Zbl 0176.51401
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