A Krasnosel’skii-Type Result for Any Non Simply Connected Orthogonal Polygon

Abstract

Let $S$ be an orthogonal polygon and let $A_1,\dots ,A_n$ represent pairwise disjoint sets, each the connected interior of an orthogonal polygon, $A_i\subseteq S,1\le i\le n$. Define $T=S\setminus (A_1\cup \dots \cup A_n)$. We have the following Krasnosel’skii-type result: Set $T$ is staircase star-shaped if and only if $S$ is staircase star-shaped and every $4n$ points of $T$ see via staircase paths in $T$ a common point of $\text{Ker }S$. Moreover, the proof offers a procedure to select a particular collection of $4n$ points of $T$ such that the subset of $\text{Ker }S$ seen by these $4n$ points is exactly $\text{Ker }T$. When $n=1$, the number 4 is best possible.