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Line Transversals to Disjoint Balls

Abstract

We prove that the set of directions of lines intersecting three disjoint balls in ℝ3 in a given order is a strictly convex subset of \(\mathbb {S}^{2}\) . We then generalize this result to n disjoint balls in ℝd. 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.

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Correspondence to Ciprian Borcea.

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Borcea, C., Goaoc, X. & Petitjean, S. Line Transversals to Disjoint Balls. Discrete Comput Geom 39, 158–173 (2008). https://doi.org/10.1007/s00454-007-9016-z

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Keywords

  • Transversal
  • Geometric permutation
  • Convexity