Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 194–212

# 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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## 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