Property $P\sb{3}$ and the union of two convex sets

Abstract

A set S in a linear space is said to have the threepoint convexity property P3 iff for each triple of points x, y, z of S, at least one of the segments xy, xz, yz is a subset of S. It is proved that if S is a compact set in Euclidean space of dimension at least three with at least one point to its kernel and if the set of points of local nonconvexity of S is to its hull, then S has property P3 iff it is the union of two sets. Introduction. Property P3 has been defined and investigated by Valentine for two-dimensional sets [3] anid in finite-dimensional spaces [5]. A set S in a linear space is said to have the three-point convexity property P3 iff for each triple of points x, y, zES, at least olne of the segments xy, xz, yz is a subset of S. Although every set which is the union of two sets has property P3, property P3 alone does not characterize such sets, as the example of a five-pointed star shows. Valentine [3] has shown that in E2 a closed set having property P3 can be expressed as the union of three or fewer sets, and that the number three is best in this case. It is shown in this paper that under certain conditions a set in Euclidean space of dimension three (or higher) is the union of two sets if and only if it has property P3. McKinney [2 ] has shown that if S is a closed set in a topological linear space, then S is the union of two sets if and only if it has the property that for any cyclically ordered n-tuple (n odd) of points of S, at least one of the segments connecting consecutive points is a subset of S. (This property implies property P3.) Marr and Stamey [1 ] also have considered a property stronger than property P3 which implies that S is the union of two sets. Preliminary definitions and results. A point x of a set S in En is called a point of local nonconvexity of S if for every neighborhood N of x there exists a pair of points u, vESnN such that the segment itv is not a subset of S. The kernel of the set S will be denoted by K, and the set of points of local nonconvexity of S will be denoted by Q. The boundary operator in En will be denoted by bd, interior by int, closure by cl, and convex hull by conv. THEOREM 1 (VALENTINE [5]). Let S be a closed connected set having Received by the editors June 18, 1969. 1 This paper comes from a doctoral dissertation with Professor F. A. Valentine at the University of California, Los Angeles, and was supported in part by NSF Grant