A Krasnosel'skij theorem for staircase paths in orthogonal polygons (Q1339779)

Let \(S\) be a simply connected orthogonal polygon in \(\mathbb{R}^ 2\). If every three points on the boundary of \(S\) are visible via staircase paths from a common point of \(S\), then all points of \(S\) are visible via staircase paths from some common point of \(S\). A similar statement is also true. If every three points of \(S\), excluding the reflex vertices, are clearly visible via staircase paths from some common point of \(S\), then all points of \(S\) are clearly visible via staircase paths from a common point of \(S\). Here, \(q\) is clearly visible from \(p\) means that any point in \(S\) and in some neighborhood of \(p\) sees all points in some neighborhood of \(q\) and in \(S\).