A deterministic polynomial time algorithm for heilbronn’s problem in dimension three

Abstract

Heilbronn conjectured that among any n points in the 2- dimensional unit square [0, 1]2, there must be three points which form a triangle of area at most O(1/n2). This conjecture was disproved by Komlós, Pintz and Szemerédi [15] who showed that for every n there exists a configuration of n points in the unit square [0, 1]2 where all triangles have area at least Ω(log n/n2). Here we will consider a 3-dimensional analogue of this problem and we will give a deterministic polynomial time algorithm which finds n points in the unit cube [0, 1]3 such that the volume of every tetrahedron among these n points is at least Ω(log n/n3).