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
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Distributions of points in the unit square and large k-gons
European Journal of Combinatorics
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
The on-line heilbronn’s triangle problem in d dimensions
COCOON'06 Proceedings of the 12th 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 in the 2-dimensional unit square [0, 1]2, 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 [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]2 where all triangles have area at least Ω(log n/n2). Considering a generalization of this problem to dimensions d≥3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least Ω(1/nd). We improve on this lower bound by a logarithmic factor Θ(log n).