Cutting dense point sets in half

A halving hyperplane of a set S of n points in R(d) contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two half-spaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most delta n(1/d), delta some constant. Such a set S is called dense. In d = 2 dimensions the number of halving lines for a dense set can be as much as Omega(n log n), and it cannot exceed O (n(5/4)/log* n). The upper bound improves over the current best bound of O (n(3/2)/log* n) which holds more generally without any density assumption. In d = 3 dimensions we show that O (n(7/3)) is an upper bound on the number of halving planes for a dense set, The proof is based on a metric argument that can be extended to d greater than or equal to 4 dimensions, where it leads to O (n(d-2/d)) as an upper bound for the number of halving hyperplanes.