A Krasnosel'skii theorem for open finitely starlike sets in \(R^ 3\) (Q2639340)

A set S in Euclidean d-space \({\mathbb{R}}^ d\) is called finitely starlike if every finite subset of S sees via S a common point. The main result of this paper says that an open set \(S\subset {\mathbb{R}}^ 3\) is finitely starlike if every seven points in S see via S a common point, and if for every three points x,y,z\(\in S\) which see each other via S the inclusion cov\(\{\) x,y,z\(\}\subseteq S\) holds true.