On Intersection of Maximal Orthogonally k-Starshaped Polygons

Let F be a simply connected orthogonal polygon in R ² and let P denote the intersection of all maximal orthogonally k-starshaped polygons in F for any fixed integer k,k2. If P and for every x,y P which are joined in F by a staircase path having two segments there is a similar staircase path from x to y in P, then there exists a maximal orthogonally k-starshaped polygon Q in F such that the staircase k-kernel of Q is a subset of the staircase k-kernel of P. In particular, F is either an orthogonally k-starshaped simply connected polygon in F or empty.