Distributions of points and large quadrangles

Authors:
Hanno Lefmann
Affiliations:
Fakultät für Informatik, TU Chemnitz, Chemnitz, Germany
Venue:
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Year:
2004

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Abstract

We consider a variant of Heilbronn's triangle problem by asking, given any integer n ≥ 4, for the supremum Δ4(n) of the minimum area determined by the convex hull of some four of n points in the unit-square [0,1]2 over all distributions of n points in [0,1]2 Improving the lower bound Δ4(n) = Ω (1/n3/2) of Schmidt [19], we will show that Δ4(n) = Ω ((log n)1/2/n3/2) as asked for in [5], by providing a deterministic polynomial time algorithm which finds n points in the unit-square [0,1]2 that achieve this lower bound.