Piercing axis-parallel boxes

Maria Chudnovsky, Sophie Spirkl, Shira Zerbib

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

Let F be a finite family of axis-parallel boxes in ℝd such that F contains no k + 1 pairwise disjoint boxes. We prove that if F contains a subfamily M of k pairwise disjoint boxes with the property that for every F ∈ F and M ∈ M with F ∩ M ≠ ∅, either F contains a corner of M or M contains 2d −1 corners of F, then F can be pierced by O(k) points. One consequence of this result is that if d = 2 and the ratio between any of the side lengths of any box is bounded by a constant, then F can be pierced by O(k) points. We further show that if for each two intersecting boxes in F a corner of one is contained in the other, then F can be pierced by at most O(k log log(k)) points, and in the special case where F contains only cubes this bound improves to O(k).

Original languageEnglish (US)
Article number#P1.70
JournalElectronic Journal of Combinatorics
Volume25
Issue number1
DOIs
StatePublished - Mar 29 2018

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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