Covering spheres with spheres
Abstract
Given a sphere of any radius $r$ in an $n$dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average number of solid spheres covering a point in a bigger sphere. For a growing dimension $n,$ we design a covering that has covering density of order $(n\ln n)/2$ for the full Euclidean space or for a sphere of any radius $r>1.$ This new upper bound reduces two times the asymptotic order of $n\ln n$ established in the classical Rogers bound.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2006
 arXiv:
 arXiv:math/0606002
 Bibcode:
 2006math......6002D
 Keywords:

 Mathematics  Metric Geometry
 EPrint:
 11 pages, 1 figure