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Tilings, packings, coverings, and the approximation of functions
Author(s) -
Hinrichs Aicke,
Richter Christian
Publication year - 2004
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310151
Subject(s) - mathematics , disjoint sets , unit (ring theory) , compact space , combinatorics , space (punctuation) , extension (predicate logic) , metric space , context (archaeology) , unit sphere , discrete mathematics , pure mathematics , paleontology , linguistics , philosophy , mathematics education , computer science , biology , programming language
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. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)