Local convexity and πΏ_{π} sets
F. A. Valentine Β· Proceedings of the American Mathematical Society Β· 1965
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.