Israel Journal of Mathematics

, Volume 190, Issue 1, pp 213–228 | Cite as

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



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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  1. [1]
    G. Ambrus, A. Bezdek and F. Fodor, A Helly-type transversal theorem for n-dimensional unit balls, Archiv der Mathematik 86 (2006), 470–480.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    C. Borcea, X. Goaoc and S. Petitjean, Line transversals to disjoint balls, Discrete & Computational Geometry 1–3 (2008), 158–173.CrossRefMathSciNetGoogle Scholar
  3. [3]
    O. Cheong, X. Goaoc, A. Holmsen and S. Petitjean, Hadwiger and Helly-type theorems for disjoint unit balls, Discrete & Computatioanl Geometry 1–3 (2008), 194–212.CrossRefMathSciNetGoogle Scholar
  4. [4]
    L. Danzer, Über ein Problem aus der kombinatorischen Geometrie, Archiv der Mathematik 8 (1957), 347–351.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    L. Danzer, B. Grünbaum and V. Klee, Helly’s theorem and its relatives, in Convexity, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, 1963, pp. 101–180.Google Scholar
  6. [6]
    J. Eckhoff, Helly, Radon and Carathéodory type theorems, in Handbook of Convex Geometry, Vol. A, North-Holland, Amsterdam, 1993, pp. 389–448.Google Scholar
  7. [7]
    X. Goaoc, Some discrete properties of the space of line transversals to disjoint balls, in Non-Linear Computational Geometry, Springer, Berlin, 2008, pp. 51–83.Google Scholar
  8. [8]
    J. E. Goodman, R. Pollack and R. Wenger, Geometric transversal theory, in New Trends in Discrete and Computational Geometry, Vol. 10, Algorithms and Combinatorics, Springer, Berlin, 1993, 163–198.CrossRefGoogle Scholar
  9. [9]
    B. Grünbaum, On common transversals, Archiv für Mathematische Logik und Grundlagenforschung 9 (1958), 465–469.MATHGoogle Scholar
  10. [10]
    H. Hadwiger, Über Eibereiche mit gemeinsamer Treffgeraden, Portugaliae Mathematica 16 (1957), 23–29.MATHMathSciNetGoogle Scholar
  11. [11]
    A. Holmsen, M. Katchalski and T. Lewis, A Helly-type theorem for line transversals to disjoint unit balls, Discrete & Computational Geometry 29 (2003), 595–602.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Holmsen and J. Matoušek, No Helly theorem for stabbing translates by lines ind, Discrete & Computational Geometry 31 (2004), 405–410.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Katchalski, A conjecture of Grünbaum on common transversals, Mathematica Scandinavica 59 (1986), 192–198.MATHMathSciNetGoogle Scholar
  14. [14]
    H. Pottmann and J. Wallner, Computational Line Geometry, Springer-Verlag, Heidelberg, Berlin, 2001.MATHGoogle Scholar
  15. [15]
    H. Tverberg, Proof of Grünbaum’s conjecture on common transversals, Discrete & Computational Geometry 4 (1989), 191–203.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    R. Wenger, Helly-type theorems and geometric transversals, in Handbook of Discrete & Computational Geometry, 2nd edition, CRC Press LLC, Boca Raton, 2004, pp. 73–96.Google Scholar

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© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceKAISTDaejeonSouth Korea
  2. 2.LORIA — INRIA Nancy Grand-Estvillers-lès-NancyFrance
  3. 3.Department of Mathematical SciencesKAISTDaejeonSouth Korea

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