Covering spheres with spheres
Ilya Dumer
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 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 n established in the classical Rogers bound.
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