Abstract
Let S subset-of R3 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(n2.998).As a main tool, for every set Y of n points in the plane a set N of size O(n4) is constructed such that the points of N are distributed almost evenly in the triangles determined by Y.
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