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Disjoint Unit Spheres admit at Most Two Line Transversals

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Algorithms - ESA 2003 (ESA 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2832))

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Abstract

We show that a set of n disjoint unit spheres in ℝd admits at most two distinct geometric permutations, or line transversals, if n is large enough. This bound is optimal.

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References

  1. Asinowski, A.: Common transversals and geometric permutations. Master’s thesis, Technion IIT, Haifa (1998)

    Google Scholar 

  2. Edelsbrunner, H., Sharir, M.: The maximum number of ways to stab n convex non-intersecting sets in the plane is 2n − 2. Discrete Comput. Geom. 5, 35–42 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Google Scholar 

  4. Holmsen, A., Katchalski, M., Lewis, T.: A Helly-type theorem for line transversals to disjoint unit balls. Discrete Comput. Geom. 29, 595–602 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Huang, Y., Xu, J., Chen, D.Z.: Geometric permutations of high dimensional spheres. In: Proc. 12th ACM-SIAM Sympos. Discrete Algorithms, pp. 244–245 (2001)

    Google Scholar 

  6. Katchalski, M., Lewis, T., Liu, A.: The different ways of stabbing disjoint convex sets. Discrete Comput. Geom. 7, 197–206 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Katchalski, M., Suri, S., Zhou, Y.: A constant bound for geometric permutations of disjoint unit balls. Discrete & Computational Geometry 29, 161–173 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Katz, M.J., Varadarajan, K.R.: A tight bound on the number of geometric permutations of convex fat objects in ℝd. Discrete Comput. Geom. 26, 543–548 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Smorodinsky, S., Mitchell, J.S.B., Sharir, M.: Sharp bounds on geometric permutations for pairwise disjoint balls in ℝd. Discrete Comput. Geom. 23, 247–259 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wenger, R.: Upper bounds on geometric permutations for convex sets. Discrete Comput. Geom. 5, 27–33 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wenger, R.: Helly-type theorems and geometric transversals. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 4, pp. 63–82. CRC Press LLC, Boca Raton (1997)

    Google Scholar 

  12. Zhou, Y., Suri, S.: Geometric permutations of balls with bounded size disparity. Comput. Geom. Theory Appl. 26, 3–20 (2003)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Otfried Cheong

  2. LORIA (INRIA Lorraine), 615, rue du Jardin Botanique, B.P. 101, 54602, Villers-les-Nancy, France

    Xavier Goaoc

  3. School of Computing, Soongsil University, Seoul, South Korea

    Hyeon-Suk Na

Authors
  1. Otfried Cheong
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  2. Xavier Goaoc
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  3. Hyeon-Suk Na
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica e Automazione, Università degli Studi “Roma Tre”, via della Vasca Navale 79, 00146, Rome, Italy

    Giuseppe Di Battista

  2. School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Uri Zwick

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© 2003 Springer-Verlag Berlin Heidelberg

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Cheong, O., Goaoc, X., Na, HS. (2003). Disjoint Unit Spheres admit at Most Two Line Transversals. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_14

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  • DOI: https://doi.org/10.1007/978-3-540-39658-1_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20064-2

  • Online ISBN: 978-3-540-39658-1

  • eBook Packages: Springer Book Archive

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