Generalizations of Heilbronn's Triangle Problem

For integers d,j⩾2 and n⩾j, distributions of n points in the d-dimensional unit cube d[0,1] are investigated, such that the minimum volume of the convex hull determined by j of n points is large. Lower and upper bounds on these minimum volumes are given. For obtaining lower bounds, results on the independence number of non-uniform, linear hypergraphs are used, which might be of interest by their own.