Covering a ball with smaller equal balls in \(\mathbb R^n\)
Let \(\nu_{T,n}\), \(T>\frac{1}{2}\), \(n \geq 2\), be the minimum number of balls of radius \(\frac{1}{2}\) which are needed to cover a ball of radius \(T\) in \(\mathbb{R}^n\). Upper bounds on \(\nu_{T,n}\) were obtained by \textit{C. A. Rogers} [Mathematika, Lond. 10, 157--164 (1963; Zbl 0158.19603)]. In the paper under review, the author improves the Rogers bound when \(T<\frac{n}{2\ln n}\). The asymptotic consequences are described and further improved by using some results of \textit{K. Börödzky, Jr.} and \textit{G. Wintsche} [Discrete Comput. Geom. 25, 237--253 (2003; Zbl 1006.52012)]. Lower bounds on \(\nu_{T,n}\) are also discussed.
scientific article; zbMATH DE number 2138100