Dual Helly-Type Theorems for Unions of Sets Star-Shaped via Staircase Paths

Abstract

Let $d$ be a fixed integer, $0\le d\le 2$, and let $\mathcal{K}$ be a family of sets in the plane having simply connected union. Assume that for every countable subfamily $\{K_n:n\ge 1\}$ of $\mathcal{K}$, the union $\cup \{K_n\ge 1\}$ is
starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least $d$. Then, $\cup \{K:K\in \mathcal{K}\}$ has these properties as well.
In the finite case ,define function $g$ on $(0,1,2)$ by $g(0)=2$, $g(1)=g(2)=4$. Let $\mathcal{K}$ be a finite family of nonempty compact sets in the plane such that $\cup \{K\in \mathcal{K}\}$ has a connected complement. For fixed $d\in \{0,1,2\}$, assume that for every $g(d)$ members of $\mathcal{K}$, the corresponding union is starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least $d$. Then, $\cup \{K\in \mathcal{K}\}$ also has these properties,also.
Most of these results are dual versions of theorems that hold for intersections of sets starshaped via staircase paths.The exceotion is the finite case above when $d=2$ .Surprisingly ,although the result for $d=2$ holds for unique of sets, no analogue for intersections of sets is possible.