# Covering spheres with spheres

Research paper by Ilya Dumer

Indexed on: 31 May '06Published on: 31 May '06Published in: Mathematics - Metric Geometry

#### Abstract

Given a sphere of radius $r>1$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with unit spheres. Our goal is to design a covering of the lowest covering density, which defines the average number of unit spheres covering a point in a bigger sphere. For growing $n,$ we obtain the covering density of $(n\ln n)/2.$ This new upper bound is half the order $n\ln n$ established in the classic Rogers bound.