On Heilbronn's problem in higher dimension
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
A Lower Bound for Heilbronn's Triangle Problem in d Dimensions
SIAM Journal on Discrete Mathematics
Convex Hulls of Point-Sets and Non-uniform Hypergraphs
AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
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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In this paper we show lower bounds for the on-line version of Heilbronn's triangle problem in three and four dimensions. Specifically, we provide incremental constructions for positioning n points in the 3-dimensional (resp., 4-dimensional) unit cube, for which every tetrahedron (resp., pentahedron) defined by four (resp., five) of these points has volume 驴( 1/n3.333... ) (resp., 驴( 1/n5.292... )).