On Unions of Two Convex Sets

Valentine (3) introduced the three-point convexity property P3 : a set S in En satisfies P3 if for each triple of points x, y, z in S at least one of the closed segments xy, yz, xz is in S. He proved, (3 or 1) that in the plane a closed connected set satisfying P3 is the union of some three convex subsets. The problem of characterizing those sets that are the union of two convex subsets was suggested. Stamey and Marr (2) have provided an answer for compact subsets of the plane. We present here a generalization of property P3 which characterizes closed sets in an arbitrary topological linear space which are the union of two convex subsets.