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Pinning a Line by Balls or Ovaloids in ℝ.

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Title: Pinning a Line by Balls or Ovaloids in ℝ.
Authors: Goaoc, Xavier1 goaoc@loria.fr, König, Stefan2 koenig@ma.tum.de, Petitjean, Sylvain1 Sylvain.Petitjean@inria.fr
Source: Discrete & Computational Geometry. Mar2011, Vol. 45 Issue 2, p303-320. 18p.
Subjects: Line geometry, Ovals, Finite geometries, Intersection theory, Permutations, Convex domains
Abstract: We show that if a line ℓ is an isolated line transversal to a finite family $\mathcal{F}$ of (possibly intersecting) balls in ℝ and no two balls are externally tangent on ℓ, then there is a subfamily $\mathcal{G}\subseteq\mathcal{F}$ of size at most 12 such that ℓ is an isolated line transversal to $\mathcal{G}$. We generalize this result to families of semialgebraic ovaloids. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: We show that if a line ℓ is an isolated line transversal to a finite family $\mathcal{F}$ of (possibly intersecting) balls in ℝ and no two balls are externally tangent on ℓ, then there is a subfamily $\mathcal{G}\subseteq\mathcal{F}$ of size at most 12 such that ℓ is an isolated line transversal to $\mathcal{G}$. We generalize this result to families of semialgebraic ovaloids. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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