Distributions of Points and Large Convex Hulls of k Points

Abstract

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]^d 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]^d, 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]^d, 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.