WALCOM 2009: WALCOM: Algorithms and Computation pp 44-46

Line Transversals and Pinning Numbers

• Otfried Cheong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5431)

Abstract

A line transversal to a family of convex objects in ℝ d is a line intersecting each member of the family. There is a rich theory of geometric transversals, see for instance the surveys of Danzer et al. [6], Eckhoff [7], Goodman et al. [8] and Wenger [11].

Keywords

Computational Geometry Line Transversal Convex Object Disjoint Ball Helly Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
Aronov, B., Cheong, O., Goaoc, X., Rote, G.: Lines pinning lines (manuscript, 2008)Google Scholar
2. 2.
Borcea, C., Goaoc, X., Petitjean, S.: Line transversals to disjoint balls. Discrete & Computational Geometry 1-3, 158–173 (2008)
3. 3.
Cheong, O., Goaoc, X., Holmsen, A.: Lower bounds for pinning lines by balls (manuscript, 2008)Google Scholar
4. 4.
Cheong, O., Goaoc, X., Holmsen, A., Petitjean, S.: Hadwiger and Helly-type theorems for disjoint unit spheres. Discrete & Computational Geometry 1-3, 194–212 (2008)
5. 5.
Danzer, L.: Über ein Problem aus der kombinatorischen Geometrie. Archiv der Mathematik (1957)Google Scholar
6. 6.
Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Klee, V. (ed.) Convexity, Proc. of Symposia in Pure Math., pp. 101–180. Amer. Math. Soc. (1963)Google Scholar
7. 7.
Eckhoff, J.: Helly, Radon and Caratheodory type theorems. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Convex Geometry, pp. 389–448. North Holland, Amsterdam (1993)
8. 8.
Goodman, J.E., Pollack, R., Wenger, R.: Geometric transversal theory. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry. Algorithms and Combinatorics, vol. 10, pp. 163–198. Springer, Heidelberg (1993)
9. 9.
Holmsen, A., Katchalski, M., Lewis, T.: A Helly-type theorem for line transversals to disjoint unit balls. Discrete & Computational Geometry 29, 595–602 (2003)
10. 10.
Holmsen, A., Matoušek, J.: No Helly theorem for stabbing translates by lines in ℝd. Discrete & Computational Geometry 31, 405–410 (2004)
11. 11.
Wenger, R.: Helly-type theorems and geometric transversals. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete & Computational Geometry, 2nd edn., ch. 4, pp. 73–96. CRC Press LLC, Boca Raton (2004)Google Scholar