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
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
Hi-index | 0.00 |
We consider a variant of Heilbronn's triangle problem by asking for fixed integers d,k ≥2 and any integer n ≥k for a distribution of n points in the d-dimensional unit cube [0,1]d such that the minimum volume of the convex hull of k points among these n points is as large as possible. We show that there exists a configuration of n points in [0,1]d, such that, simultaneously for j = 2, ..., k, the volume of the convex hull of any j points among these n points is Ω( 1/n(j−−1)/(1+|d−−j+1|)). Moreover, for fixed k ≥d+1 we provide a deterministic polynomial time algorithm, which finds for any integer n ≥k a configuration of n points in [0,1]d, which achieves, simultaneously for j = d+1, ..., k, the lower bound Ω( 1/n(j−−1)/(1+|d−−j+1|)) on the minimum volume of the convex hull of any j among the n points.