Ln Sets and the Closures of Open Connected Sets
Nick M. Stavrakas · Canadian Journal of Mathematics · 1975
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.