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On the number of halving planes

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Abstract

LetS ⊂ℝ3 be ann-set in general position. A plane containing three of the points is called a halving plane if it dissectsS into two parts of equal cardinality. It is proved that the number of halving planes is at mostO(n 2.998).

As a main tool, for every setY ofn points in the plane a setN of sizeO(n 4) is constructed such that the points ofN are distributed almost evenly in the triangles determined byY.

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Authors and Affiliations

  1. Hungarian Academy of Sciences, P.O.B. 127, 1364, Budapest, Hungary

    I. Bárány & Z. Füredi

  2. Department of Computer Science, Eötvös Loránd University, Múzeum krt. 6.-8., 1088, Budapest, Hungary

    L. Lovász

  3. Princeton University, 08544, Princeton, NJ, USA

    L. Lovász

Authors
  1. I. Bárány
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  2. Z. Füredi
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  3. L. Lovász
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Additional information

Research supported partly by the Hungarian National Foundation for Scientific Research grant No. 1812

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Bárány, I., Füredi, Z. & Lovász, L. On the number of halving planes. Combinatorica 10, 175–183 (1990). https://doi.org/10.1007/BF02123008

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  • DOI: https://doi.org/10.1007/BF02123008

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