Covering a ball with smaller equal balls in ~$\rb^{n}$

Jean-Louis Verger-Gaugry (1)

1 IF - Institut Fourier

We give an explicit upper bound of the minimal number ~$\nu_{T,n}$~ of balls of radius $1/2$ which form a covering of a ball of radius ~$T > 1/2$~ in ~$\rb^{n}, n \geq 2$. The asymptotic estimates of ~$\nu_{T,n}$~ 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 ~$\rb^{n}$~ covering the sphere ~$\sbb^{n-1}$. The optimality of the asymptotic estimates is discussed.

Mots clés: sphere covering

Domaines: Théorie des nombres [math.NT]

Jean-Louis Verger-Gaugry. Covering a ball with smaller equal balls in ~$\rb^{n}$. Discrete and Computational Geometry, 2005, 33, pp.143--155. ⟨hal-00346015⟩