A Krasnosel'skij-type theorem in the plane involving polygonal \(n\)-paths (Q1331266)

This short note shows one theorem: If \(S\) is a compact simply connected subset of \(R^ 2\), then \(\text{ker}_ n(S)\) is nonempty if and only if every three boundary points of \(S\) are visible via polygonal \(n\)-paths in \(S\) from a common point. Here we say that a point \(x\) is visible from a point \(y\) via a polygonal \(n\)-path in \(S\) (a non-empty subset of the Euclidean plane) if and only if there is in \(S\) a polygonal path of at most \(n\) closed line segments joining \(x\) to \(j\). The set of points of \(S\) which see all points of \(S\) via polygonal \(n\)-paths in \(S\) is called the \(n\)-th order kernel of \(S\), denoted by \(\text{ker}_ n(S)\).