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  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  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 .
- Grid Point
- Convex Hull
- Minimum Area
- London Mathematical Society
- Simple Geometric Argument
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