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
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
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In this paper it is shown that for any set of n points selected from the d-dimensional unit cube, d odd, the volume of the smallest simplex spanned by the set is $O(n^{-(1+{1\over 2d})})$, which is a slight improvement on the only known upper bound O(n-1)$, although still far from the lower bound $\Omega(n^{-d}\log n)$.