A Helly Theorem For Intersections of Sets Starshaped via Staircase \(n\)-Paths

Abstract

For \(n \geq 1\), let \(p(n)\) denote the smallest natural number \(r\) for which the following is true: For \(K\) any finite family of simply connected orthogonal polygons in the plane and points \(x\) and \(y\) in \(\cap\{K : K \in \mathcal{K}\}\), if every \(r\) (not necessarily distinct) members of \(K\) contain a common staircase \(n\)-path from \(x\) to \(y\), then \(\cap\{K : K \in \mathcal{K}\}\) contains such a staircase path. It is proved that \(p(1) = 1, p(2) = 2, p(3) = 4, p(4) = 6\), and \(p(n) \leq 4 + 2p(n – 2)\) for \(n \geq 5\).

The numbers \(p(n)\) are used to establish the following result. For \(\mathcal{K}\) any finite family of simply connected orthogonal polygons in the plane, if every \(3p(n + 1)\) (not necessarily distinct) members of \(\mathcal{K}\) have an intersection which is starshaped via staircase \(n\)-paths, then \(\cap\{K : K \in \mathcal{K}\}\) is starshaped via staircase \((n+1)\)-paths. If \(n = 1\), a stronger result holds.