Convex polytopes in restricted point sets in $\mathbb{R}^d$

Abstract

For a finite point set P ⊂ R d , denote by diam(P ) the ratio of the largest to the smallest distances between pairs of points in P . Let c d,α (n) be the largest integer c such that any n-point set P ⊂ R d in general position, satisfying diam(P ) < α d √ n, contains an c-point convex independent subset. We determine the asymptotics of c d,α (n) as n → ∞ by showing the existence of positive constants β = β(d, α) and γ = γ(d) such that βn d−1 d+1 ≤ c d,α (n) ≤ γn d−1 d+1 for α ≥ 2.