Generating a staircase starshaped set and its kernel in $${{\mathbb R}^3}$$ from certain staircase convex subsets

Abstract

Let \({\mathcal{C} = \{C_{1}, \ldots, C_{n}\}}\) be a family of distinct boxes in \({\mathbb{R}^3}\) whose intersection graph is a tree, and let \({S = C_{1}\cup \cdots \cup C_n.}\) Assume that S is starshaped via staircase paths with corresponding staircase kernel K, and let A denote the smallest box containing K. Then K is staircase convex and \({S \cap A = K}\) . When S ≠ K, for every point p in S\K define Wp = {s : s lies on some staircase path in S from p to a point of K }. Each set Wp is staircase convex. Further, there is a minimal collection \({\mathcal{W}}\) of Wp sets whose union is S. The collection \({\mathcal{W}}\) is unique and finite, and \({\mathcal{W}}\) is exactly the collection of maximal Wp sets in S. Finally, \({\cap \{ W : W\, {\rm in}\, \mathcal{W} \}}\) is exactly the kernel K.