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Helly-Type Theorems for Line Transversals to Disjoint Unit Balls.

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Title: Helly-Type Theorems for Line Transversals to Disjoint Unit Balls.
Authors: Otfried Cheong1, Xavier Goaoc2, Andreas Holmsen3, Sylvain Petitjean4
Source: Discrete & Computational Geometry. Mar2008, Vol. 39 Issue 1-3, p194-212. 19p. 2 Diagrams.
Subjects: Integral theorems, Integrals, Integral calculus, Fractional integrals
Abstract: 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≥2d disjoint unit balls in ℝ d 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 ℝ d admits a line transversal if any subfamily of size 4d−1 admits a transversal. [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: Helly-Type Theorems for Line Transversals to Disjoint Unit Balls.
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  Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Mar2008, Vol. 39 Issue 1-3, p194-212. 19p. 2 Diagrams.
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  Data: 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≥2d disjoint unit balls in ℝ d 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 ℝ d admits a line transversal if any subfamily of size 4d−1 admits a transversal. [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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      – SubjectFull: Integral calculus
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