A deterministic polynomial-time algorithm for Heilbronn's problem in three dimensions

Heilbronn conjectured that among arbitrary n points in the two-dimensional unit square [0, 1](2), there must be three points which form a triangle of area O(1/n(2)). This conjecture was disproved by a nonconstructive argument of Komlos, Pintz, and Szemeredi [J. London Math. Soc., 25 (1982), pp. 13-24], 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 Omega(log n/n(2)). Here we will consider a three-dimensional analogue of this problem and show how to find deterministically in polynomial time n points in the unit cub [0, 1](3) such that the volume of very tetrahedron among these n points is Omega(log n/n(3)).