A Helly theorem for intersections of orthogonally starshaped sets (Q1413640)

A nonempty set in the plane is called an orthogonal polygon if it is a connected union of finitely many rectangles whose edges are parallel to the coordinate axes. By a staircase path we mean a path which is a union of successive ``horizontal'' and ``vertical'' segments. Set \(S\) is called starshaped via staircase paths if there exists a point \(p \in S\) such that for every point \(r \in S\) there is a staircase path in \(S\) which contains \(p\) and \(r\). The main theorem says the following. Let \(\mathcal F\) be a finite family of simply connected orthogonal polygons in the plane. If every three of them have nonempty intersection which is starshaped via staircase paths, then the intersection of all sets from \(\mathcal F\) is a simply connected orthogonal polygon which is starshaped via staircase paths.