Distributions of Points in d Dimensions and Large k-Point Simplices

Abstract

We consider a variant of Heilbronn’s triangle problem by asking for fixed dimension d ≥ 2 and for fixed integers k ≥ 3 with k ≤ d+1 for a distribution of n points in the d-dimensional unit-cube [0,1]^d such that the minimum volume of a k-point simplex among these n points is as large as possible. Denoting by Δ _k,d (n) the supremum of the minimum volume of a k-point simplex among n points over all distributions of n points in [0,1]^d we will show that c _k . (log n)^{1/( d − − k + 2)}/n ^{(k − − 1)/(d − − k + 2)} ≤ Δ _k,d (n) ≤ c _k ′/n ^{(k − − 1)/d} for 3 ≤ k ≤ d +1, and moreover Δ _k,d (n) ≤ c _k ′′/n ^{(k − − 1)/d + (k − − 2)/(2d(d− − 1))} for k ≥ 4 even, and constants c _k , c _k ′, c _k ′′ > 0.