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2005 On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings
G. Muraz, J. -L. Verger-Gaugry
Experiment. Math. 14(1): 47-57 (2005).

Abstract

We study lower bounds of the packing density of a system of nonoverlapping equal spheres in $\rb^{n}, n \geq 2,$ as a function of the maximal circumradius of its Voronoi cells. Our viewpoint, using Delone sets, allows us to investigate the gap between the upper bounds of Rogers or Kabatjanskii-Levenstein and the Minkowski-Hlawka type lower bounds for the density of lattice-packings, without entering the fundamental problem of constructing Delone sets with Delone constants between $2^{-0.401}$ and $1$. As a consequence we provide explicit asymptotic lower bounds of the covering radii (holes) of the Barnes-Wall, Craig, and Mordell-Weil lattices, respectively $BW_{n},$ $\ab_{n}^{(r)},$ and $MW_{n}$, and of the Delone constants of the BCH packings, when $n$ goes to infinity.

Citation

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G. Muraz. J. -L. Verger-Gaugry. "On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings." Experiment. Math. 14 (1) 47 - 57, 2005.

Information

Published: 2005
First available in Project Euclid: 30 June 2005

zbMATH: 1108.52021
MathSciNet: MR2146518

Subjects:
Primary: 52C17 , 52C23

Keywords: Delone set , Density , hole , sphere packing

Rights: Copyright © 2005 A K Peters, Ltd.

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