Clear visibility and sets which are almost starshaped

Abstract

Let S S be a nonempty compact connected set in R 2 {R^2} . Assume that every two points of local nonconvexity of S S are clearly visible from a common point of S S . Then S S is "almost" starshaped. In fact, for any point y y in R 2 {R^2} there is a line L L through y y such that L ∩ S L \cap S is convex and every point of S S sees via S S a point of L L . If "clearly visible" is replaced by the weaker term "visible", then the result fails.