Suitable families of boxes and kernels of staircase starshaped sets in $${\mathbb{R}^d}$$ R d

Abstract

Let S be an orthogonal polytope in \({\mathbb{R}^d}\) . There exists a suitable family \({\mathcal{C}}\) of boxes with \({S = \cup \{C : C {\rm in} \mathcal{C}\}}\) such that the following properties hold: The staircase kernel Ker S is a union of boxes in \({\mathcal{C}}\). Let \({\mathcal{V}}\) be the family of vertices of boxes in \({\mathcal{C}}\) , and let \({v_o\, \epsilon \mathcal{V}}\) . Point vo belongs to Ker S if and only if vo sees via staircase paths in S every point w in \({\mathcal{V}}\) . Moreover, these staircase paths may be selected to consist of edges of boxes in \({\mathcal{C}}\). Let B be a box in \({\mathcal{C}}\) with vertices of B in Ker S. Box B lies in Ker S if and only if, for some b in relintB and for every translate H of a coordinate hyperplane at \({b, b \epsilon}\) Ker (H ∩ S). For point p in S, p belongs to Ker S if and only if, for every x in S, there exist some p − x geodesic λ (p, x) and some corresponding \({\mathcal{C}}\) - chain D containing λ (p, x) such that D is staircase starshaped at p.