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.