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Covering and packing properties of bounded sequences of convex sets

Published online by Cambridge University Press:  26 February 2010

H. Groemer
H. Groemer
Affiliation:
Department of Mathematics, The University of Arizona, Tucson, Arizona, U.S.A., AZ 85721.
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In this paper n always denotes an arbitrary but fixed positive integer. Let S be a subset of n-dimensional euclidean space En and (Si) = (S1, S2, …) a finite or infinite sequence of subsets of En. The sequence (Si) is called a covering of S if S ⊂ ⋃Si, and a packing in S if ⋃Si ⊂ S and int Si, ⋂ int Sj = Ø (for all i ≠ j). We say that (Si) permits an isometric covering of S or packing in S if there are rigid motions σi so that (σiSi) is a covering of S or a packing in S, respectively. If there are not only rigid motions but translations τi so that (τiSi) is a covering or packing, we express this by saying that (Si) permits a translative covering or packing. We consider sequences (Si) rather than sets {Si} not because the ordering is of any importance but because some of the sets Si may appear repeatedly.

Type
Research Article
Information
Mathematika , Volume 29 , Issue 1 , June 1982 , pp. 18 - 31
Copyright
Copyright © University College London 1982

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