Hadwiger and Helly-type theorems for disjoint unit spheres
<|reference_start|>Hadwiger and Helly-type theorems for disjoint unit spheres: We prove Helly-type theorems for line transversals to disjoint unit balls in $\R^{d}$. In particular, we show that a family of $n \geq 2d$ disjoint unit balls in $\R^d$ has a line transversal if, for some ordering $\prec$ of the balls, any subfamily of 2d balls admits a line transversal consistent with $\prec$. We also prove that a family of $n \geq 4d-1$ disjoint unit balls in $\R^d$ admits a line transversal if any subfamily of size $4d-1$ admits a transversal.<|reference_end|>
@article{cheong2007hadwiger,
    title={Hadwiger and Helly-type theorems for disjoint unit spheres},
    author={Otfried Cheong, Xavier Goaoc (INRIA Lorraine - LORIA), Andreas
  Holmsen, Sylvain Petitjean (INRIA Lorraine - LORIA)},
    journal={Discrete and Computational Geometry (2006)},
    year={2007},
    archivePrefix={arXiv},
    eprint={cs/0702039},
    primaryClass={cs.CG}
}