Let \(S\) be a simple polygon in the plane whose vertices may be partitioned into sets \(A’, B’\), such that for every two points of \(A’\) (of \(B’\)), the corresponding segment is in \(S\). Then \(S\) is a union of \(6\) (or possibly fewer) convex sets. The number \(6\) is best possible. Moreover, the simple connectedness requirement for set \(S\) cannot be removed.