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Israel Journal of Mathematics

, Volume 190, Issue 1, pp 213–228 | Cite as

Lower bounds to helly numbers of line transversals to disjoint congruent balls

  • Otfried Cheong
  • Xavier Goaoc
  • Andreas Holmsen
Article

Abstract

A line is a transversal to a family F of convex objects in ℝ d if it intersects every member of F. In this paper we show that for every integer d ⩾ 3 there exists a family of 2d−1 pairwise disjoint unit balls in ℝ d with the property that every subfamily of size 2d − 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Crucial to the proof is the notion of a pinned transversal, which means an isolated point in the space of transversals. Here we investigate minimal pinning configurations and construct a family F of 2d−1 disjoint unit balls in ℝ d with the following properties: (i) The space of transversals to F is a single point and (ii) the space of transversals to any proper subfamily of F is a connected set with non-empty interior.

Keywords

Line Transversal Empty Interior Disjoint Ball HELLY Number Common Transversal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceKAISTDaejeonSouth Korea
  2. 2.LORIA — INRIA Nancy Grand-Estvillers-lès-NancyFrance
  3. 3.Department of Mathematical SciencesKAISTDaejeonSouth Korea

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