On the number of halving planes

@article{Brny1990OnTN,
  title={On the number of halving planes},
  author={Imre B{\'a}r{\'a}ny and Zolt{\'a}n F{\"u}redi and L{\'a}szl{\'o} Lov{\'a}sz},
  journal={Combinatorica},
  year={1990},
  volume={10},
  pages={175-183}
}
Let S ⊂ IR be an n-set in general position. A plane containing three of the points is called a halving plane if it dissects S into two parts of equal cardinality. It is proved that the number of halving planes is at most O(n). As a main tool, for every set Y of n points in the plane a set N of size O(n) is constructed such that the points of N are distributed almost evenly in the triangles determined by Y .