A visible shoreline theorem for staircase paths (Q621374)

The author proves the following Theorem: ``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 \(P\subseteq \mathbb{R}^{2}\setminus (\text{int }C)\). If every four points of \(P\) see a common boundary point of \(C\) via staircase paths in \(\mathbb{R}^{2}\setminus (\text{int }C)\), then there is a boundary point \(b\) of \(C\) such that every point of \(P\) sees \(b\) (via staircase paths in \(\mathbb{R}^{2}\setminus (\text{int }C)\)). The number four is the best possible, even if \(C\) is orthogonally convex.'' Few results referring to the orthogonal convex hull of a set \(C\) are obtained in order to prove the main theorem. First of all, a manner of generating the orthogonal convex hull of a set from the inside of \(C\) is described. Few visibility properties of the points belonging to the set difference between the orthogonal convex hull of a set and itself are also given.