On intersection of maximal orthogonally \(k\)-starshaped polygons (Q1818718)

A nonempty set \(S\in \mathbb R^2\) is called an orthogonal polygon if \(S\) is a connected union of finitely many convex polygons whose edges are parallel to the coordinate axes. Let \(\lambda\) be a simple polygonal path whose edges are parallel to coordinate axes. Such a path \(\lambda\) is called a staircase path if the associated vectors alternate in direction. If the staircase path \(\lambda\) is a union of at most \(k\) edges, then \(\lambda\) is a staircase \(k\)-path. A set \(S\) is called staircase \(k\)-convex provided for every \(x, y\) in \(S\), \(x\) sees \(y\) via staircase \(k\)-paths. Similarly, \(S\) is starshaped via staircase \(k\)-paths (orthogonally \(k\)-starshaped) if for some point \(p\) in \(S,\) \(p\) sees each point of \(S\) via staircase \(k\)-paths, and the set of all such points \(p\) is the staircase \(k\)-kernel of \(S,\) denoted Ker\(_k S.\) A subset \(F\) of \(S\) is called maximal orthogonally \(k\)-starshaped polygon if \(F\) is orthogonally \(k\)-starshaped polygon and if \(F\) is not a proper subset of any other orthogonally \(k\)-starshaped polygon of \(S.\) In this paper the author investigates the problem of describing the intersection of maximally starshaped via staircase \(k\)-paths polygons in \(\mathbb R^2.\) Let \(F\) be a simply connected orthogonal polygon in \(\mathbb R^2\) and \(\mathcal F=\{\mathcal F_\alpha:\alpha\in I\}\) denote the family of all maximal orthogonally \(k\)-starshaped polygons in \(F\) for any fixed integer \(k, k\geqslant 2.\) Let \(P=\cap_{\alpha\in I}\mathcal F_\alpha\neq\varnothing\) and for every \(a,b\in P\) joined by a staircase path having two segments in \(F\) there is a similar staircase path from \(a\) to \(b\) in \(P.\) Then there exists \(\mathcal Q\in\mathcal F\) such that Ker\(_k \mathcal Q\subseteq\) Ker\(_k P.\) In particular, \(P\) is either an orthogonally \(k\)-starshaped simply connected polygon in \(F\) or empty.