Decompositions for nonclosed planarm-convex sets

Abstract

1Φ Introduction* For S a subset of Euclidean space, S is said to be m-convex, m ^ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in S. Several decomposition theorems have been proved for m-convex sets in the plane. A closed planar 3-convex set is expressible as a union of 3 or fewer convex sets (Valentine [4]), and an arbitrary planar 3-convex set is a union of 6 or fewer convex sets (Breen [1]). Concerning the general case, a recent study shows that for m ^ 3, a closed planar m-convex set may be decomposed into (m — 1)2~ or fewer convex sets (Kay and Breen [2]). This leads naturally to the problem considered here, that of determining whether such a bound exists for an arbitrary m-convex set SQR: With the restriction that (int cl S) ~ S contain no isolated points, a bound in terms of m is obtained; without this restriction, an example reveals that no bound is possible.