Critical visibility and outward rays (Q1110098)

This paper is a contribution to the geometry of starshaped sets. Let S be a set in a locally convex space. If \(x\in S\), the star of x in S (denoted st(x,S)) is the set \(\{\) \(y\in S:\) [x,y]\(\subset S\}\); ker S (the kernel of S) is, as usually, the (convex) set \(\{x\in S:st(x,S)=S\}\). If \(int S\neq \emptyset,\) S is closed and \(x\in S\), let \(cv(x,S)=int S\cap \partial (st(x,S))\) (the set of critical visibility of x in S). If \(y\in lc(S)\) and \(p\in cv(y,S)\), then (p,y)\(\cap \ln c(S)\neq \emptyset\) (here lc(S) and lnc(S) denote the set of points of local convexity and local nonconvexity of S, resp.). If S is regular (i.e. \(S=\overline{int S}\)) and nonconvex, then \(\partial (st(x,S))\cap \ln c(S)\neq \emptyset\) for each \(x\in S\). The kernel of S is expressed as the intersection of certain subsets (the inner stems of x, where x runs over lnc(S)). In the planar case, the author also obtains theorems of Krasnoselskij-type.