-partitions and a characterization for compact unions of starshaped sets

Author:
Marilyn Breen

Journal:
Proc. Amer. Math. Soc. **102** (1988), 677-680

MSC:
Primary 52A30; Secondary 52A35

MathSciNet review:
929001

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Abstract | References | Similar Articles | Additional Information

Abstract: For natural numbers and , define , and otherwise. The set in has property if and only if is a finite union of one-dimensional convex sets and for every -member subset of there correspond points (depending on ) such that each point of sees via some . The following results are established.

(1) Let and be fixed natural numbers, and let be a collection of sets such that every members meet in at most one point. Then has a -partition with if and only if every or fewer members of have such a -partition.

(2) Let be compact in . The set is a union of starshaped sets if and only if there is a sequence of compact sets converging to (relative to the Hausdorff metric) such that each set satisfies property .

The first result yields a piercing number for certain families of sets, while the second provides a characterization for compact unions of starshaped sets in .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0929001-6

Keywords:
-partitions,
unions of starshaped sets,
Krasnosel'skii-type theorems

Article copyright:
© Copyright 1988
American Mathematical Society