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Circles through two points that always enclose many points


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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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).

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  • Main Ingredient
  • Counting Formula
  • Edge Counting