Distributions of Points and Large Convex Hulls of k Points

We consider a variant of Heilbronn’s triangle problem by asking for fixed integers d, k ≥ 2 and any integer n ≥ k for a distribution of n points in the d-dimensional unit cube [0, 1] such that the minimum volume of the convex hull of k points among these n points is as large as possible. We show that there exists a configuration of n points in [0, 1], 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, for fixed k ≥ d+1 we provide a deterministic polynomial time algorithm, which finds for any integer n ≥ k a configuration of n points in [0, 1], which achieves, simultaneously for j = d + 1, . . . , k, the lower bound Ω(1/n(j−1)/(1+|d−j+1|)) on the minimum volume of the convex hull of any j among the n points.