Point Sets in the Unit Square and Large Areas of Convex Hulls of Subsets of Points

Abstract

In this paper generalizations of Heilbronn's triangle problem are considered. By using results on the independence number of linear hypergraphs, for fixed integers k ≥ 3 and any integers n ≥ k a o(n6k-4) time deterministic algorithm is given, which finds distributions of n points in the unit square [0, 1]2 such that, simultaneously for j = 3, ..., k, the areas of the convex hulls determined by any j of these n points are Ω((log n)1/(j-2)/n(j-1)/(j-2)).