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
Distributions of points and large quadrangles
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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
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We consider a generalization of Heilbronn's triangle problem by asking, given any integers n ≥ k ≥ 3, for the supremum Δk(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 Δk(n) = Ω(1/n(k-1)/(k-2}) from [5] and from [20] for k = 4, we will show that Δk(n) = Ω((log n)1/(k-2)/n(k-1)/(k-2) for each fixed integer k ≥ 3 as asked for in [5]. We will also provide a deterministic polynomial time algorithm which finds n points in the unit-square [0, 1]2 achieving this lower bound.