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arXiv · cs/0702039

Hadwiger and Helly-type theorems for disjoint unit spheres

Otfried Cheong·Xavier Goaoc·Andreas Holmsen·Sylvain Petitjean
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Abstract

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.

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Otfried Cheong, Xavier Goaoc, Andreas Holmsen, Sylvain Petitjean. 2007-02-07. Hadwiger and Helly-type theorems for disjoint unit spheres. https://arxiv.org/abs/cs/0702039

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