Improved Bounds for Intersecting Triangles and Halving Planes

If a configuration of m triangles in the plane has only n points as vertices, then there must be a set of max { dm/(2n− 5)e Ω(m3/(n6 log n)) triangles having a common intersection. As a consequence the number of halving planes for a three-dimensional point set is O(n log n). For all m and n there exist configurations of triangles in which the largest common intersection involves max {dm/(2n− 5)e O(m2/n3) triangles; the upper and lower bounds match for m = O(n). The best previous bounds were Ω(m/(n log n)) for intersecting triangles, and O(n log n) for halving planes.