For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ 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 $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < α\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, α}(n)$ as $n \to \infty$ by showing the existence of positive constants $β= β(d, α)$ and $γ= γ(d)$ such that $βn^{\frac{d-1}{d+1}} \le c_{d, α}(n) \le γn^{\frac{d-1}{d+1}}$ for $α\geq 2$.