Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 194–212 | Cite as

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

  • Otfried Cheong
  • Xavier Goaoc
  • Andreas Holmsen
  • Sylvain Petitjean
Article

Abstract

We prove Helly-type theorems for line transversals to disjoint unit balls in ℝd. In particular, we show that a family of n≥2d disjoint unit balls in ℝd has a line transversal if, for some ordering of the balls, any subfamily of 2d balls admits a line transversal consistent with . We also prove that a family of n≥4d−1 disjoint unit balls in ℝd admits a line transversal if any subfamily of size 4d−1 admits a transversal.

Keywords

Geometric transversal theory Helly-type theorem Hadwiger-type theorem Spheres Balls Line transversal 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Otfried Cheong
    • 1
  • Xavier Goaoc
    • 2
  • Andreas Holmsen
    • 3
  • Sylvain Petitjean
    • 4
  1. 1.Division of Computer ScienceKAISTDaejeonSouth Korea
  2. 2.LORIA–INRIA LorraineNancyFrance
  3. 3.Department of MathematicsUniversity of BergenBergenNorway
  4. 4.LORIA–CNRSNancyFrance

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