Line Transversals to Disjoint Balls

We prove that the set of directions of lines intersecting three disjoint balls in ℝ 3 in a given order is a strictly convex subset of . We then generalize this result to n disjoint balls in ℝ d . As a consequence, we can improve upon several old and new results on line transversals to disjoint balls...

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Bibliographic Details
Published in:Discrete & computational geometry 2008-03, Vol.39 (1-3), p.158-173
Main Authors: Borcea, Ciprian, Goaoc, Xavier, Petitjean, Sylvain
Format: Article
Language:English
Subjects:
Combinatorics
Computational Geometry
Computational Mathematics and Numerical Analysis
Computer Science
Geometry
Mathematics
Mathematics and Statistics
Set theory
Theorems
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Summary:We prove that the set of directions of lines intersecting three disjoint balls in ℝ 3 in a given order is a strictly convex subset of . We then generalize this result to n disjoint balls in ℝ d . As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-007-9016-z