-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

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

MathSciNet review:
929001

Full-text PDF

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 .

**[1]**Marilyn Breen,*A characterization theorem for compact unions of two starshaped sets in 𝑅³*, Pacific J. Math.**128**(1987), no. 1, 63–72. MR**883377****[2]**Marilyn Breen,*A Krasnosel′skiĭ-type theorem for unions of two starshaped sets in the plane*, Pacific J. Math.**120**(1985), no. 1, 19–31. MR**808926****[3]**Marilyn Breen,*Clear visibility and unions of two starshaped sets in the plane*, Pacific J. Math.**115**(1984), no. 2, 267–275. MR**765187****[4]**Marilyn Breen,*Sets with convex closure which are unions of two starshaped sets and families of segments which have a 2-partition*, J. Geom.**27**(1986), no. 1, 1–23. MR**858238**, https://doi.org/10.1007/BF01230330**[5]**Ludwig Danzer and Branko Grünbaum,*Intersection properties of boxes in 𝑅^{𝑑}*, Combinatorica**2**(1982), no. 3, 237–246. MR**698651**, https://doi.org/10.1007/BF02579232**[6]**Ludwig Danzer, Branko Grünbaum, and Victor Klee,*Helly's theorem and its relatives*, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R.I., 1963, pp. 101-180.**[7]**M. Krasnosselsky,*Sur un critère pour qu’un domaine soit étoilé*, Rec. Math. [Mat. Sbornik] N. S.**19(61)**(1946), 309–310 (Russian, with French summary). MR**0020248****[8]**Steven R. Lay,*Convex sets and their applications*, John Wiley & Sons, Inc., New York, 1982. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR**655598****[9]**Sam B. Nadler Jr.,*Hyperspaces of sets*, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions; Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49. MR**0500811****[10]**Frederick A. Valentine,*Convex sets*, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR**0170264**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
52A30,
52A35

Retrieve articles in all journals with MSC: 52A30, 52A35

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