Point sets in the unit square and large areas of convex hulls of subsets of points

In this paper generalizations of Heilbronn’s triangle problem to convex hulls of j points in the unit square 0,1]² are considered. By using results on the independence number of linear hypergraphs, for fixed integers k≥3 and any integers n≥k a deterministic o(n ^6k−4) time algorithm is given, which finds distributions of n points in 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)}).