Clear visibility and $L\sb 2$ sets

Let S C R2 be a closed connected set whose points of local nonconvexity are compact. Suppose any two points of local nonconvexity are clearly visible from a common point of S. Then S is almost starshaped and S is 2-polygonally connected. This generalizes a result of Breen. We begin with some definitions. Let S C R2 and let x, y E S. We say x sees y via S if [x, y] c S. Let A c S be convex. Then S is almost starshaped with respect to A if every point of S sees some point of A via S. We say S is an L2 set provided given x, y E S there exists a polygonal path in S joining x to y consisting of at most two closed line segments. Let 1 be a polygonal arc joining x to y in S. Then 1 is called a minimum 2-path if x does not see y via S, 1 is a polygonal arc consisting of two closed line segments and if 1' is any other 2-path joining x to y in S, the arclength of 1 is less than or equal to the arclength of 1'. We say x is clearly visible from y if there exists a neighborhood N of x such that y sees each point of N n s via S. Finally, int S and Q denote the interior and points of local nonconvexity of S, respectively. M. Breen in [1], has shown that if S is a compact, connected set in R2 such that any two points of Q are clearly visible from a common point of S, then S is almost starshaped. We show that her theorem holds with the weaker hypothesis that S be closed and Q be compact and show that S is an L2 set. We prove the following generalized version of Theorem 1 of [1]. THEOREM 1. Let S c R2 be closed and connected. Suppose Q is compact and any two points of Q are clearly visible from a common point of S. Then given any point y E R2 there is a line 1 through y such that 1 n S is convex and S is almost starshaped with respect to 1 n s. Furthermore, S is an L2 set. PROOF. For each (x, y) E Q x Q, our hypotheses imply the existence of a point Z(x,y) and open sets NX and Ny such that Z(x,y) sees NX n S and Ny n s via S. Let A = {(Nx,Ny)I(x,y) E Q x Q}. Since Q x Q is compact we may select a finite subcover {(Nxl, Nyl), (Nx2 XNY2). .. I (NxnlNYn)}. Let zi = Z(x,,y,). Let K be a compact convex subset of R2 such that (Q U {z1, Z2, . .X Zn}) C int K. Note Kn s is a polygonally connected set since given w, t E K n s there exist by Lemma 1 of [3] points qw, qt E Q such that [w, qw] c KnS and [t, qt] C KnS. Hence for some i we must have that [w, qw] U [qw, zi] U [z,, qt] U [qt, t] c S. Thus K n s is a compact, connected set whose points of local nonconvexity are exactly Q. Note Kn s satisfies the hypotheses of Theorem 1 of Breen [1]. Thus given any p E R2 there exists a line lk such that Ip = lk nKns is convex and such Received by the editors April 1, 1986 and, in revised form, July 24, 1987. 1980 Mathematics Subject Classification (1985 Revi'sion). Primary 52A30, 52A35. (?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page