Helly-type theorems for infinite and for finite intersections of sets starshaped via staircase paths

Marilyn Breen

Abstract: 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

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