  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 $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point
set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) <
\alpha\sqrt[d]{n}$, contains an $c$-point convex independent subset. We
determine the asymptotics of $c_{d, \alpha}(n)$ as $n \to \infty$ by showing
the existence of positive constants $\beta = \beta(d, \alpha)$ and $\gamma =
\gamma(d)$ such that $\beta n^{\frac{d-1}{d+1}} \le c_{d, \alpha}(n) \le \gamma
n^{\frac{d-1}{d+1}}$ for $\alpha\geq 2$.
