A helly number for unions of two boxes inR2

LetSbe a polygonal region in the plane with edges parallel to the coordinate axes. If every5or fewer boundary points ofScan be partitioned into setsAandBso thatconv A⋃ conv B⫅S, thenSis a union of two convex sets, each a rectangle. The number5is best possible.Without suitable hypothesis on edges ofS, 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.