Generalized convexity: CP3 and boundaries of convex sets

A set S is convex if for every pair of points P, Q~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 P3: 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 P3, it is a union of at most three convex sets. In this paper we study a property closely related to, but weaker than, P3' We say that S has property CP3 ("collinear P3") if P3 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 a. 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. Convexity Convex arcs Convex curves Generalized convexity Boundaries l. INTRODUCTION A set S is convex if for every pair of points P, QeS, the line segment PQ is contained in S. This definition can be generalized in various ways. For example, (1) S is called starshapedfrom Po E S if PoQ is contained in S for all QeS; thus S is convex if it is starshaped from all of its points. As another example, (2) S is called ortho-convex if PQ is contained is S for all P, Q ~ S such that PQ is horizontal or vertical. One class of generalizations of convexity, due to Valentine, 13) makes use of triples (or k-tuples), rather than pairs, of points. A set satisfies Valentine's property P3 if 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. For example, a polygonal arc consisting of two non-collinear line segments (Fig. 1) is not convex, but is easily seen to have property P3. (Note that the three-segment polygonal arc in Fig. lb does not even have property P3-) It can be shown that if a set has property P3, it is a union of at most three convex sets. In this paper …