Generalizations of Heilbronn’s triangle problem

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