A visible shoreline theorem for staircase paths

Abstract

Let C be an orthogonal polygon in the plane, bounded by a simple closed curve, and assume that C is starshaped via staircase paths. Let \(P \subseteq {\mathbb{R}}^2\backslash({\rm int} C)\). If every four points of P see a common boundary point of C via staircase paths in \({\mathbb{R}}^2\backslash({\rm int} C)\), then there is a boundary point b of C such that every point of P sees b (via staircase paths in \({\mathbb{R}}^2\backslash({\rm int} C)\)). The number four is best possible, even if C is orthogonally convex.