Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 49, No. 2, pp. 527539 (2008) 

Hellytype theorems for infinite and for finite intersections of sets starshaped via staircase pathsMarilyn BreenUniversity of Oklahoma, Norman, Oklahoma 73019, U.S.A., email: mbreen@ou.eduAbstract: Let $d$ be a fixed integer, $0 \leq d \leq 2$, and let $\K$ be a family of simply connected sets in the plane. For every countable subfamily $\{K_n : n \geq 1 \}$ of $\K$, assume that $\cap \{K_n : n \geq 1 \}$ is starshaped via staircase paths and that its staircase kernel contains a convex set of dimension at least $d$. Then $\cap \{K : K$ in $\K \}$ has these properties as well. For the finite case, define function $f$ on $\{ 0, 1 \}$ by $f (0) = 3, f (1) = 4$. Let $\K = \{K_i : 1 \leq i \leq n \}$ be a finite family of compact sets in the plane, each having connected complement. For $d$ fixed, $d\, \epsilon\, \{0, 1 \}$, and for every $f (d)$ members of $\K$, assume that the corresponding intersection is starshaped via staircase paths and that its staircase kernel contains a convex set of dimension at least $d$. Then $\cap \{K_i : 1 \leq i \leq n \}$ has these properties, also. There is no analogous Helly number for the case in which $d = 2$. Each of the results above is best possible. Keywords: Sets starshaped via staircase paths Classification (MSC2000): 52.A30, 52.A35 Full text of the article:
Electronic version published on: 18 Sep 2008. This page was last modified: 28 Jan 2013.
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