Some Krasnosel'skij numbers for finitely starlike sets in the plane (Q1106473)

Let x and y be points of a subset Q of Euclidean plane \({\mathbb{R}}^ 2\). We say that x sees y via Q if the segment connecting x and y lies in Q. A set \(S\subset {\mathbb{R}}^ 2\) is called finitely starlike if for every finite \(F\subset S\) there exists \(p\in S\) such that every point of F sees p via S. The author proves the following Theorem. Let \(S\subset {\mathbb{R}}^ 2\) and \((int cl S)\sim S=\emptyset.\) Then S is finitely starlike if and only if every four points of S see a common point via S.