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
Distributions of points and large quadrangles
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Generalizations of Heilbronn's triangle problem
European Journal of Combinatorics
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We consider a generalization of Heilbronn's triangle problem by asking, given any integers n=k, for the supremum @D"k(n) of the minimum area determined by the convex hull of some k of n points in the unit square [0,1]^2, where the supremum is taken over all distributions of n points in [0,1]^2. Improving the lower bound @D"k(n)=@W(1/n^(^k^-^1^)^/^(^k^-^2^)) from [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn's problem, SIAM Journal on Computing 30 (2000) 383-390] and from [W.M. Schmidt, On a problem of Heilbronn, Journal of the London Mathematical Society (2) 4 (1972) 545-550] for k=4, we show that @D"k(n)=@W((logn)^1^/^(^k^-^2^)/n^(^k^-^1^)^/^(^k^-^2^)) for fixed integers k=3 as asked for in [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn's problem, SIAM Journal on Computing 30 (2000) 383-390]. Moreover, we provide a deterministic polynomial time algorithm which finds n points in [0,1]^2, which achieve this lower bound on @D"k(n).