LOCAL CONVEXITY AND Ln SETS1

We will prove that a closed connected set S in ET whose points of local nonconvexity can be decomposed into a finite number of convex parts also possesses a simple type of polygonal connectedness. Furthermore, if S has a finite number of points of local nonconvexity, we obtain another result. In order to describe this situation easily we use the notation of Ln set used by Horn and Valentine [l ] in 1949. For other results related to Ln sets, see A. Bruckner and J. Bruckner [l] and Valentine [l]. The following notations and definitions will enable us to express the results more readily. Notations. The interior, boundary and closure of a set S in Euclidean r-space Er are denoted by int S, bd 5 and cl 5 respectively. If xES, yES, then xy denotes the closed line segment joining x and y. The symbols U, C\ and ~ denote set union, set intersection and set difference respectively. The symbol 0 is used to denote the empty set. The convex hull of a set SEE, is indicated by conv S. Definition 1. A set 5 is called an Ln set if each pair of points in S can be joined by a polygonal arc of S containing at most n segments. Definition 2. A point x E S is a point of local convexity of 5 if there exists a neighborhood N of x such that Nf~\S is convex; otherwise, x is called a point of local nonconvexity of S. Definition 3. A set S is starshaped relative to a point p if for every x E S it is true that pxES. The following two theorems together with Theorem 3 stated later contain the main results of this paper. The symbols n and r always denote non-negative integers.