An algorithm for Heilbronn's problem

Abstract

Heilbronn conjectured that given arbitrary n points from R ², 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 Komlós, Pintz and Szemerédi [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.

We then consider a generalization of this problem as introduced by Schmidt [10] to convex hulls of k points. We obtain the following result: For every k ≥ 4, there is a polynomial-time algorithm which on input n computes n points in the unit square such that the convex hull of any k points has area at least Ω(1/n^(k-i)/(k-2)). For k = 4, the existence of such a configuration has been proved in [10].

This research was supported by the Deutsche Forschungsgemeinschaft as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).