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Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

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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.

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Author information

Correspondence to Xavier Goaoc.

Additional information

Andreas Holmsen was supported by the Research Council of Norway, prosjektnummer 166618/V30. Otfried Cheong and Xavier Goaoc acknowledge support from the French-Korean Science and Technology Amicable Relationships program (STAR).

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Cheong, O., Goaoc, X., Holmsen, A. et al. Helly-Type Theorems for Line Transversals to Disjoint Unit Balls. Discrete Comput Geom 39, 194–212 (2008). https://doi.org/10.1007/s00454-007-9022-1

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Keywords

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