Covering a ball with smaller equal balls in ℝn

Abstract

We give an explicit upper bound of the minimal number νT,n of balls of radius 1/2 which form a covering of a ball of radius T > 1/2 in ℝn, n ≥ 2. The asymptotic estimates of νT,n we deduce when n is large are improved further by recent results of Böröczky, Jr. and Wintsche on the asymptotic estimates of the minimal number of equal balls of ℝn covering the sphere double-struck S signn-1. The optimality of the asymptotic estimates is discussed.