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

Let \( S \) be an orthogonal polygon and let \( A_1, \ldots, A_n \) represent pairwise disjoint sets, each the connected interior of an orthogonal polygon, \( A_i \subseteq S, 1 \leq i \leq n \). Define \( T = S \setminus (A_1 \cup \ldots \cup A_n) \). 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.