The points of local nonconvexity of starshaped sets

Abstract

of S arbitrarily close to K, unless S itself be convex. Klee's generalization of the just quoted Tietze's theorem follows immediately. The notion of higher visibility is introduced in the last section, and three Erasnosselsky-type theorems involving the points of local nonconvexity are proved. 1* Notations and basic definitions* The interior, closure, boundary and convex hull of a set S are denoted by int S, clS, bdry S and conv S, respectively. The closed segment joining x and y is denoted [xy]. If xeS and yeS, we say that x sees y via S if [xy] c S. The star of x with respect to S is the set st(x; S) of all points of S that see x via S. A star-center of S is a point xeS such that st(x; S) = S, that is a point of S that sees the whole S. The kernel of S is the set ker S of all the star-centers of S. S is star shaped if ker S Φ 0. A convex component of S is a maximal convex subset of S. The point x e bdry S is a point of local nonconvexity of S if for every neighborhood U of x, the set U' = U Π S is not convex. The set of all points of local nonconvexity of S is denoted Inc S. The origin (null-vector) of a linear topological space is denoted by θ, and the family of its neighborhoods by