A Helly Theorem For Intersections of Sets Starshaped via Staircase $n$-Paths

Abstract

For $n\ge 1$, 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 $p(n)\le 4+2p(n–2)$ for $n\ge 5$.

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.