An Algorithm for Heilbronn's Problem

Abstract

Heilbronn conjectured that given arbitrary n points from R2, located in the unit square (or disc), there must be three points which form a triangle of area at most O(1/n2). This conjecture was shown to be false by a nonconstructive argument of Komlos, Pintz and Szemeredi [6] who showed that for every n there is a configuration of n points in the unit square where all triangles have area at least Ω(log n/n2). In this paper, we provide a polynomial-time algorithm which for every n computes such a configuration of n points.