The average-case area of Heilbronn-type triangles
Random Structures & Algorithms
A Deterministic Polynomial Time Algorithm for Heilbronn's Problem in Dimension Three
LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
Distributions of points in the unit-square and large k-gons
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Large triangles in the d-dimensional unit cube
Theoretical Computer Science - Computing and combinatorics
Distributions of points in the unit square and large k-gons
European Journal of Combinatorics
Convex Hulls of Point-Sets and Non-uniform Hypergraphs
AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
Generalizations of Heilbronn's triangle problem
European Journal of Combinatorics
Point sets in the unit square and large areas of convex hulls of subsets of points
COCOA'07 Proceedings of the 1st international conference on Combinatorial optimization and applications
Distributions of points and large convex hulls of k points
AAIM'06 Proceedings of the Second international conference on Algorithmic Aspects in Information and Management
Distributions of points in d dimensions and large k-point simplices
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Distributions of points and large quadrangles
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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Heilbronn conjectured that given arbitrary n points from the 2-dimensional unit square, there must be three points which form a triangle of area at most O(1/n2). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz, and Szemerédi [ J. London Math. Soc., 25 (1982), pp. 13--24] who showed that for every n there is a configuration of n points in the unit square where all triangles have area at least $\Omega({\log n}/{n^2})$. Considering a discretization of Heilbronn's problem, we give an alternative proof of the result from [J. London Math. Soc., 25 (1982), pp. 13--24]. Our approach has two advantages: First, it yields a polynomial-time algorithm which for every n computes a configuration of n points where all triangles have area $\Omega({\log n}/{n^2})$. Second, it allows us to consider a generalization of Heilbronn's problem to convex hulls of k points where we can show that an algorithmic solution is also available.