Cutting dense point sets in half

Abstract

A halving hyperplane of a set S of n points in ℝd 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 δn1/d, δ 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 Ω, (n log n) , and it cannot exceed O (n5/4/log* n). The upper bound improves over the current best bound of O(n3/2/log* n) which holds more generally without any density assumption. In d = 3 dimensions we show that O(n7/3) 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 ≥ 4 dimensions, where it leads to O(nd-2/d) as an upper bound for the number of halving hyperplanes.