A Krasnosel'skii theorem for orthogonal polygons starshaped via staircase \((n + 1)\)-paths
Krasnosel'skii showed that a non-empty compact set \(S\) in the plane is starshaped if and only if every three points of \(S\) are visible (along straight line segments in \(S\)) from a common point. The author considers Krasnosel'skii-type results for the case where \(S\) is assumed to be a simply connected orthogonal polygon (orthogonality refers to axis parallel sides), and straight line segments are replaced by staircase paths. In an earlier paper she had shown that a simply connected orthogonal polygon \(S\) is starshaped via staircase paths if and only if every two points of \(S\) are visible via staircase paths in \(S\) from a common point. The present work refines this statement: If \(n\geq1\) is such that every two points of \(S\) are visible via staircase paths of length \(n\) in \(S\) from a common point, then \(S\) is starshaped via staircase paths of length \(n+1\). It is shown that the number two is best possible and that \(n+1\) is best possible for \(n\geq 2\). Moreover, the associated staircase \((n+1)\)-kernel is shown to be staircase path convex with paths of lengths at most \(n+1\).
scientific article; zbMATH DE number 5851435