Random Graphs 93 Proceedings of the sixth international seminar on Random graphs and probabilistic methods in combinatorics and computer science
The Algorithmic Aspects of Uncrowded Hypergraphs
SIAM Journal on Computing
A Deterministic Polynomial-Time Algorithm for Heilbronn's Problem in Three Dimensions
SIAM Journal on Computing
An Algorithm for Heilbronn's Problem
SIAM Journal on Computing
A Lower Bound for Heilbronn's Triangle Problem in d Dimensions
SIAM Journal on Discrete Mathematics
The average-case area of Heilbronn-type triangles
Random Structures & Algorithms
On Heilbronn’s Problem in Higher Dimension
Combinatorica
An Upper Bound for the d-Dimensional Analogue of Heilbronn's Triangle Problem
SIAM Journal on Discrete Mathematics
Generalizations of Heilbronn's triangle problem
European Journal of Combinatorics
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We consider a variant of Heilbronn's triangle problem by asking for a distribution of n points in the d-dimensional unit cube [0, 1]d such that the minimum (two-dimensional) area of a triangle among these n points is as large as possible. Denoting by Δdoff-line (n) and Δdon-line (n) the supremum of the minimum area of a triangle among n points over all distributions of n points in [0, 1]d for the off-line and the on-line situation, respectively, for fixed dimension d ≥ 2 we show that c1ċ(log n)1/(d-1)/n2/(d-1) ≤ Δdoff-line(n) ≤ c'1/n2/d and c2/n2/(d-1) ≤ Δdon-line (n) ≤ c'2/n2/d for constants c1, c2, c'1,c'2 0 which depend on d only. Moreover, we provide a deterministic polynomial time algorithm that achieves the lower bound Ω((log n)1/(d-1))/n2/(d-1)) on the minimum area of a triangle among n points in [0, 1]d in the off-line case.