An Algorithm for Heilbronn's Problem

Heilbronn conjectured that given arbitrary n points from R, located in the unit square (or circle), there must be three points which form a triangle of area at most O(1=n). This conjecture was proved false by a nonconstructive argument of Koml os, Pintz and Szemer edi [KPS] who showed that there is a con guration of n points in the unit square where all triangles have area at least (logn=n). In this paper, we provide a polynomial-time algorithm which for every n computes such a con guration of points. We then consider a generalization of this problem as introduced by Schmidt [Sc] to convex hulls of k 4 points. We obtain the following result: For every k, 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 (1=n 1)=(k ). For k = 4, the existence of such a con guration has been proved in [Sc].