Generating a staircase starshaped set from a minimal collection of its subsets

Abstract

Let S be a simply connected orthogonal polygon in the plane. Assume that S is starshaped via staircase paths with corresponding staircase kernel K, where K ≠ S. For every point x in \({S \backslash K}\), define Wx = {s : s lies on some staircase path in S from x to a point of K}. Then there is a minimal collection \({\mathcal{W}}\) of Wx sets whose union is S. Moreover, \({\mathcal{W}}\) is unique and finite. Finally \({\mathcal{W}}\) is exactly the collection of maximal Wx sets in S.