Aequationes mathematicae

, Volume 87, Issue 1–2, pp 43–52 | Cite as

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

  • Marilyn Breen


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 


Orthogonal polytopes Staircase paths Staircase starshaped sets 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.The University of OklahomaNormanUSA

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