Finding Points in Convex Position in Density-Restricted Sets

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest integer such that any set $A$ of $n$ points in general position in $\mathbb{R}^d$, satisfying $\Delta(A) \leq \alpha n^{1/d}$ for a fixed $\alpha>0$, contains at least $\gamma_d(n)$ points in convex position. About $30$ years ago, Valtr proved that $\gamma_2(n)=\Theta(n^{1/3})$. Since then no further results have been obtained in higher dimensions. Here we continue this line of research in three dimensions and prove that $ \gamma_3(n) =\Theta(n^{1/2})$. The lower bound implies the following approximation: Given any $n$-element point set $A\subset \mathbb{R}^3$ in general position, satisfying $\Delta(A) \leq \alpha n^{1/3}$ for a fixed $\alpha$, a $\Omega(n^{-1/6})$-factor approximation of the maximum-size convex subset of points can be computed by a randomized algorithm in $O(n \log{n})$ expected time.