Families of planar sets having starlike union (Q1299934)

A polygonal path consisting of \(k\) segments is called a \(k\)-path. The main result is as follows. Let \(F\) be a finite family of compact sets in the plane, and \(k\) be a fixed natural number. If every three members of \(F\) have a union which is simply connected and starshaped via \(k\)-paths, then all members of \(F\) have a common point and their union is starshaped via \(k\)-paths.