A Helly theorem for intersections of orthogonally starshaped sets

Abstract

Let $\cal{F}$ be a finite family of simply connected orthogonal polygons in the plane. If every three (not necessarily distinct) members of $\cal{F}$ have a nonempty intersection which is starshaped via staircase paths, then the intersection $\cap \{F : F\; \hbox{in}\; \cal{F}\}$ is a (nonempty) simply connected orthogonal polygon which is starshaped via staircase paths. Moreover, the number three is best possible, even with the additional requirement that the intersection in question be nonempty. The result fails without the simple connectedness condition.