Aequationes mathematicae

, Volume 87, Issue 1–2, pp 43–52

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

Article

## 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 v o belongs to Ker S if and only if v o 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 rel int B and for every translate H of a coordinate hyperplane at $${b, b \epsilon}$$ Ker (HS).

For point p in S, p belongs to Ker S if and only if, for every x in S, there exist some px geodesic λ (p, x) and some corresponding $${\mathcal{C}}$$ - chain D containing λ (p, x) such that D is staircase starshaped at p.

## Mathematics Subject Classification

Primary 52.A30 52.A35

## Keywords

Orthogonal polytopes Staircase paths Staircase starshaped sets

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