Tilings, packings, coverings, and the approximation of functions

Abstract

A packing (resp. covering) F of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of F. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls.Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real-valued functions that rests on so-called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non-compact spaces.