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
The On-Line Heilbronn's Triangle Problem in Three and Four Dimensions
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
On Heilbronn’s Problem in Higher Dimension
Combinatorica
Distributions of points in the unit-square and large k-gons
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
An Upper Bound for the d-Dimensional Analogue of Heilbronn's Triangle Problem
SIAM Journal on Discrete Mathematics
Random Structures & Algorithms
Convex Hulls of Point-Sets and Non-uniform Hypergraphs
AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
The on-line heilbronn’s triangle problem in d dimensions
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
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We consider a variant of Heilbronn’s triangle problem by asking for fixed dimension d ≥ 2 and for fixed integers k ≥ 3 with k ≤ d+1 for a distribution of n points in the d-dimensional unit-cube [0,1]d such that the minimum volume of a k-point simplex among these n points is as large as possible. Denoting by Δk,d(n) the supremum of the minimum volume of a k-point simplex among n points over all distributions of n points in [0,1]d we will show that ck . (log n)1/( d−−k+2)/n(k−−1)/(d−−k+2) ≤ Δk,d(n) ≤ ck′/n(k−−1)/d for 3 ≤ k ≤ d +1, and moreover Δk,d(n) ≤ ck′′/n(k−−1)/d+(k−−2)/(2d(d−−1)) for k ≥ 4 even, and constants ck, ck′, ck′′ 0.