Staircase \(k\)-kernels for orthogonal polygons (Q811663)

Let \(S\neq\emptyset\) be a finite union of boxes in \(\mathbb{R}^ 2\) whose edges are parallel to the coordinate axes. If \(S\) is simply connected and starshaped via staircase paths, then the staircase kernel of \(S\), \(\hbox{Ker} S\), as the intersection of all maximal orthogonally convex polygons in \(S\), and \(\hbox{Ker} S\) is an orthogonally convex region. In general, when \(S\) is starshaped via staircase paths, then each component of \(\hbox{Ker} S\) is an orthogonally convex polygon. However, there may be maximal orthogonally convex polygons in \(S\) which fail to contain \(\hbox{Ker} S\).