Tilings, packings, coverings, and the approximation of functions

Preprint series: 02-05, Reports on Analysis

Preprint series: , Reports on Algebra and Geometry and Analysis

MSC:
52C17 Packing and covering in $n$ dimensions, See also {05B40,
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
52C22 Tilings in $n$ dimensions, See also {05B45, 51M20}
46B20 Geometry and structure of normed linear spaces

Abstract: A packing (resp. covering) $\mathcal F$ 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 $\mathcal F$. 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.

Keywords: tiling, completely saturated packing, completely reduced covering, entropy number, partition of unity, approximation of real-valued functions