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$ 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
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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?)

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