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
Generalizations of Heilbronn's triangle problem
European Journal of Combinatorics
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 among arbitrary n points in the two-dimensional unit square [0,1]2, there must be three points which form a triangle of area 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 exists a configuration of n points in the unit square [0,1]2 where all triangles have area $\Omega({\log n}/{n^2})$. Here we will consider a three-dimensional analogue of this problem and show how to find deterministically in polynomial time n points in the unit cube [0,1]3 such that the volume of every tetrahedron among these n points is $\Omega(\log n/n^3)$.