Generalizations of Heilbronnʹs Triangle Problem

For integers d , j ⩾ 2 and n ⩾ j , distributions of n points in the d-dimensional unit cube [ 0 , 1 ] d 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.