A Helly number for unions of two boxes in \(\mathbb R^2\) (Q1077724)

Summary: Let \(S\) be a polygonal region in the plane with edges parallel to the coordinate axes. If every 5 or fewer boundary points of \(S\) can be partitioned into sets \(A\) and \(B\) so that \(\mathrm{conv}\, A\cup \mathrm{conv}\, b\subseteq S\), then \(S\) is a union of two convex sets, each a rectangle. The number 5 is best possible. Without suitable hypothesis on edges of \(S\), the theorem fails. Moreover, an example reveals that there is no finite Helly number which characterizes arbitrary unions of two convex sets, even for polygonal regions in the plane.