A Decomposition Theorem for m-Convex Sets in Rd

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of S, at least one of the line segments determined by these points lies in S. A point x in S is said to be a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z Є N ⌒ S, then [y, z] ⊆ S. If S fails to be locally convex at some point a in S, then q is called a point of local nonconvexity (lnc point) of S.

Intersections of m-Convex Sets

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of line segments determined by these points lies in S. A point x in S is called a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z ∈ N⋂ S, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S.

Points of Local Nonconvexity and Finite Unions of Convex Sets

Let 5 be a subset of Rd. A point x in 5 is a point of local convexity of S if and only if there is some neighborhood U of x such that, if y, z ϵ 5 ⌒ U, then [y, z] ⊆ S. If S fails to be locally convex at some point q in 5, then q is called a point of local nonconvexity (lnc point) of S.Several interesting properties are known about sets whose lnc points Q may be decomposed into n convex sets. For S closed, connected, S ∼ Q connected, and Q having cardinality n, Guay and Kay [2] have proved that S is expressible as a union of n + 1 or fewer closed convex sets (and their result is valid in a locally convex topological vector space).

Ln Sets and the Closures of Open Connected Sets

F. A. Valentine in [4] proved the following two theorems.THEOREM 1. Let S be a closed connected subset of Rd which has at most n points of local nonconvexity. Then S is an Ln+i set.THEOREM 2. Let S be a closed connected subset of Rd whose points of local nonconvexity are decomposable into n closed convex sets. Then S is an L2n+i set.These results have been extended by a number of authors, but always with stronger hypothesis. (See [1] and [2].) Using a minimal arc technique, new pr∞fs of Theorems 1 and 2 were given in [3].Valentine remarks in [4] that Theorem 2 might be improved in the case that 5 is the closure of an open connected set. The goal of this paper is to give such an improvement for sets satisfying a particular local connectivity property.