On the number of halving planes

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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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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AMS subject classification (1980)

  • 52 A 37