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 Omega((log n)(1/(j-2))/n((j-1)/(j-2))).
heilbronn's triangle problem hypergraphs approximation algorithm