Distributions of Points and Large Convex Hulls of k Points

Abstract

AbstractWe 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.KeywordsConvex HullMinimum VolumeUnit CubeSIAM JournalLondon Mathematical SocietyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.