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 W _p = {s : s lies on some staircase path in S from p to a point of K }. Each set W _p is staircase convex. Further, there is a minimal collection ${\mathcal{W}}$ of W _p sets whose union is S. The collection ${\mathcal{W}}$ is unique and finite, and ${\mathcal{W}}$ is exactly the collection of maximal W _p sets in S. Finally, ${\cap \{ W : W\, {\rm in}\, \mathcal{W} \}}$ is exactly the kernel K.