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

Abstract

Let \(d\) be a fixed integer, \(0 \leq d \leq 2\), and let \(\mathcal{K}\) be a family of sets in the plane having simply connected union. Assume that for every countable subfamily \(\{K_n : n \geq 1\}\) of \(\mathcal{K}\), the union \(\cup\{K_n \geq 1\}\) 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.