Points and triangles in the plane and halving planes in space
Boris S. Aronov, Bernard Chazelle, Herbert Edelsbrunner, Leonidas Guibas, Micha Sharir, Rephael Wenger · 1990
We prove that for any set S of n points in the plane and n3-α triangles spanned by the points of S there exists a point (not necessarily of S) contained in at least n3-3α/(512 log5 n) of the triangles. This implies that any set of n points in three-dimensional space defines at most 6.4n8/3 log5/3 n halving planes.