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Circles through two points that always enclose many points

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Abstract

This paper proves that any set of n points in the plane contains two points such that any circle through those two points encloses at least \(n\left( {1/2 - 1\sqrt {12} } \right) + O(1) \approx n/4 \cdot 7\) points of the set. The main ingredients used in the proof of this result are edge counting formulas for k-order Voronoi diagrams and a lower bound on the minimum number of semispaces of size at most k.

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Authors and Affiliations

  1. Dept of Computer Science, University of Illinois, 61801, Urbana, Illinois, USA

    H. Edelsbrunner & N. Hasan

  2. Dept of Electrical Engineering and Computer Science, University of California, 94720, Berkeley, California, USA

    R. Seidel

  3. Dept of Computer Science, University of Illinois, 61801, Urbana, Illinois, USA

    X. J. Shen

  4. Dept of Computer Science, East China Institute of Technology, Nanjing, China

    X. J. Shen

Authors
  1. H. Edelsbrunner
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  2. N. Hasan
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  3. R. Seidel
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  4. X. J. Shen
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Additional information

Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under Grant CCR-8714565, by the second author has been partially supported by the Digital Equipment Corporation, by the fourth author has been partially supported by the Office of Naval Research under Grant N00014-86K-0416.

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Edelsbrunner, H., Hasan, N., Seidel, R. et al. Circles through two points that always enclose many points. Geom Dedicata 32, 1–12 (1989). https://doi.org/10.1007/BF00181432

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  • DOI: https://doi.org/10.1007/BF00181432

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