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Discrete & Computational Geometry

, Volume 54, Issue 3, pp 637–646 | Cite as

A Variant of the Hadwiger–Debrunner (pq)-Problem in the Plane

  • Sathish Govindarajan
  • Gabriel Nivasch
Article

Abstract

Let X be a convex curve in the plane (say, the unit circle), and let \({\mathcal {S}}\) be a family of planar convex bodies such that every two of them meet at a point of X. Then \({\mathcal {S}}\) has a transversal \(N\subset {\mathbb {R}}^2\) of size at most \(1.75\times 10^9\). Suppose instead that \({\mathcal {S}}\) only satisfies the following “(p, 2)-condition”: Among every p elements of \({\mathcal {S}}\), there are two that meet at a common point of X. Then \({\mathcal {S}}\) has a transversal of size \(O(p^8)\). For comparison, the best known bound for the Hadwiger–Debrunner (pq)-problem in the plane, with \(q=3\), is \(O(p^6)\). Our result generalizes appropriately for \({\mathbb {R}}^d\) if \(X\subset {\mathbb {R}}^d\) is, for example, the moment curve.

Keywords

Convex set Transversal Hadwiger–Debrunner (\(p, q\))-problem Weak epsilon-net Helly’s theorem Fractional Helly 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Indian Institute of ScienceBangaloreIndia
  2. 2.Ariel UniversityArielIsrael

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