A Helly theorem for an enlarged class of orthogonally starshaped sets in \(\mathbb {R}^d\)

Original Paper
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Abstract

Consider a finite family \(\mathcal {C}\) of distinct boxes in \(\mathbb {R}^d\), with G the intersection graph of \(\mathcal {C}\) and with \(S \equiv \cup \{C : C \, \hbox {in} \, \mathcal {C}\}\). Assume that G is connected and, for each block B of G, assume that the corresponding members of \(\mathcal {C}\) have a staircase convex union D(B). When this occurs, we say that the orthogonal polytope S has suitable properties. Now let \(\mathbf{S}\) be a finite family of orthogonal polytopes such that, for every nonempty subfamily \(\mathbf{S}^{'}\) of \(\mathbf{S}\), the corresponding intersection \(S^{'} \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S}^{'} \}\) (if nonempty) has suitable properties and preserves transitions between certain D(B) sets. If every \(d +1\) (not necessarily distinct) members of \(\mathbf{S}\) meet in a (nonempty) staircase starshaped set, then \(S_0 \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S} \}\) is nonempty and staircase starshaped as well.

Keywords

Orthogonal polytope Staircase starshaped set Helly theorem 

Mathematics Subject Classification

Primary 52A30 52A35 

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Copyright information

© The Managing Editors 2018

Authors and Affiliations

  1. 1.The University of OklahomaNormanUSA

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