Lower bounds to helly numbers of line transversals to disjoint congruent balls

Abstract

A line is a transversal to a family F of convex objects in ℝd if it intersects every member of F. In this paper we show that for every integer d ⩾ 3 there exists a family of 2d−1 pairwise disjoint unit balls in ℝd with the property that every subfamily of size 2d − 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Crucial to the proof is the notion of a pinned transversal, which means an isolated point in the space of transversals. Here we investigate minimal pinning configurations and construct a family F of 2d−1 disjoint unit balls in ℝd with the following properties: (i) The space of transversals to F is a single point and (ii) the space of transversals to any proper subfamily of F is a connected set with non-empty interior.

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Correspondence to Otfried Cheong.

Additional information

O. C. was supported by the Korea Science and Engineering Foundation Grant R01-2008-000-11607-0 funded by the Korean government.

The cooperation by O. C. and X. G. was supported by the INRIA Equipe Associ ée KI.

A. H. was supported by the Brain Korea 21 Project, the School of Information Technology, KAIST, in 2008.

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Cheong, O., Goaoc, X. & Holmsen, A. Lower bounds to helly numbers of line transversals to disjoint congruent balls. Isr. J. Math. 190, 213–228 (2012). https://doi.org/10.1007/s11856-011-0179-1

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Keywords

  • Line Transversal
  • Empty Interior
  • Disjoint Ball
  • HELLY Number
  • Common Transversal