A line ℓ 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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G. Ambrus, A. Bezdek and F. Fodor, A Helly-type transversal theorem for n-dimensional unit balls, Archiv der Mathematik 86 (2006), 470–480.
C. Borcea, X. Goaoc and S. Petitjean, Line transversals to disjoint balls, Discrete & Computational Geometry 1–3 (2008), 158–173.
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.
L. Danzer, Über ein Problem aus der kombinatorischen Geometrie, Archiv der Mathematik 8 (1957), 347–351.
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.
J. Eckhoff, Helly, Radon and Carathéodory type theorems, in Handbook of Convex Geometry, Vol. A, North-Holland, Amsterdam, 1993, pp. 389–448.
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.
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.
B. Grünbaum, On common transversals, Archiv für Mathematische Logik und Grundlagenforschung 9 (1958), 465–469.
H. Hadwiger, Über Eibereiche mit gemeinsamer Treffgeraden, Portugaliae Mathematica 16 (1957), 23–29.
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.
A. Holmsen and J. Matoušek, No Helly theorem for stabbing translates by lines in ℝd, Discrete & Computational Geometry 31 (2004), 405–410.
M. Katchalski, A conjecture of Grünbaum on common transversals, Mathematica Scandinavica 59 (1986), 192–198.
H. Pottmann and J. Wallner, Computational Line Geometry, Springer-Verlag, Heidelberg, Berlin, 2001.
H. Tverberg, Proof of Grünbaum’s conjecture on common transversals, Discrete & Computational Geometry 4 (1989), 191–203.
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.
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
- Line Transversal
- Empty Interior
- Disjoint Ball
- HELLY Number
- Common Transversal