Generalized convexity: CP3 and boundaries of convex sets

Abstract A set S is convex if for every pair of points P , Q ϵ S , the line segment PQ is contained in S . This definition can be generalized in various ways. One class of generalizations makes use of k -tuples, rather than pairs, of points—for example, Valentine's property P 3 : For every triple of points P , Q , R of S , at least one of the line segments PQ , QR , or RP is contained in S . It can be shown that if a set has property P 3 , it is a union of at most three convex sets. In this paper we study a property closely related to, but weaker than, P 3 . We say that S has property CP 3 (“collinear P 3 ”) if P 3 holds for all collinear triples of points of S . We prove that a closed curve is the boundary of a convex set, and a simple arc is part of the boundary of a convex set, iff they have property CP 3 . This result appears to be the first simple characterization of the boundaries of convex sets; it solves a problem studied over 30 years ago by Menger and Valentine.