Staircase Visibility and Krasnosel’skii-Type Results for Planar Compact Sets

Abstract

Some Krasnotel’skii-type results previously established for a simply connected orthogonal polygon may be extended to a nonempty compact planar set $S$ having connected complement. In particular, if every two points of $S$ are visible via staircase paths from a common point of $S$, then $S$ is starshaped via staircase paths. For $n$ fixed, $n\ge 1$, if every two points of $S$ are visible via staircase $n$-paths from a common point of $S$, then $S$ is starshaped via staircase $(n+1)$-paths. In each case, the associated staircase kernel is orthogonally convex.