Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$ k$-partitions and a characterization for compact unions of $ k$ starshaped sets

Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 102 (1988), 677-680
MSC: Primary 52A30; Secondary 52A35
MathSciNet review: 929001
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For natural numbers $ k$ and $ j$, define $ \sigma \left( {1,j} \right) = j + 2,\;\sigma \left( {2,1} \right) = 6$, and $ \sigma \left( {k,j} \right) = {k^2}j + 1$ otherwise. The set $ S$ in $ {R^d}$ has property $ A\left( k \right)$ if and only if $ S$ is a finite union of one-dimensional convex sets and for every $ \sigma \left( {k,1} \right)$-member subset $ F$ of $ S$ there correspond points $ {c_1}, \ldots ,{c_k}$ (depending on $ F$) such that each point of $ F$ sees via $ S$ some $ {c_i},1 \leqslant i \leqslant k$. The following results are established.

(1) Let $ k$ and $ j$ be fixed natural numbers, and let $ \mathcal{L}$ be a collection of sets such that every $ j + 1$ members meet in at most one point. Then $ \mathcal{L}$ has a $ k$-partition $ {\mathcal{L}_1}, \ldots ,{\mathcal{L}_k}$ with $ \cap \left\{ {L:L\;{\text{in}}\;{\mathcal{L}_i}} \right\} \ne \emptyset $ if and only if every $ \sigma \left( {k,j} \right)$ or fewer members of $ \mathcal{L}$ have such a $ k$-partition.

(2) Let $ S$ be compact in $ {R^d}$. The set $ S$ is a union of $ k$ starshaped sets if and only if there is a sequence of compact sets $ \left\{ {{S_i}} \right\}$ converging to $ S$ (relative to the Hausdorff metric) such that each set $ {S_i}$ satisfies property $ A\left( k \right)$.

The first result yields a piercing number for certain families of sets, while the second provides a characterization for compact unions of $ k$ starshaped sets in $ {R^d}$.

References [Enhancements On Off] (What's this?)

  • [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,
  • [5] Ludwig Danzer and Branko Grünbaum, Intersection properties of boxes in 𝑅^{𝑑}, Combinatorica 2 (1982), no. 3, 237–246. MR 698651,
  • [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

Similar Articles

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

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

Additional Information

Keywords: $ K$-partitions, unions of starshaped sets, Krasnosel'skii-type theorems
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society