Hadwiger and Helly-type theorems for disjoint unit spheres

Details Hadwiger and Helly-type theorems for disjoint unit spheres Hadwiger and Helly-type theorems for disjoint unit spheres

2007-02-07

arxiv.org/abs/cs/0702039v1

Discrete and Computational Geometry (2006)

Computer Science

Computational Geometry

Scientific paper

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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