Generalizations of Heilbronn’s triangle problem

AbstractFor given integers d,j≥2 and any positive integers n, distributions of n points in the d-dimensional unit cube [0,1]d are investigated, where the minimum volume of the convex hull determined by j of these n points is large. In particular, for fixed integers d,k≥2 the existence of a configuration of n points in [0,1]d is shown, such that, simultaneously for j=2,…,k, the volume of the convex hull of any j points among these n points is Ω(1/n(j−1)/(1+|d−j+1|)). Moreover, a deterministic algorithm is given achieving this lower bound, provided that d+1≤j≤k