Convex polytopes in restricted point sets in $\mathbb{R}^d$
Boris Bukh, Zichao Dong
TL;DR
This work resolves the asymptotic order of the largest guaranteed convex independent subset under a diameter constraint in $\mathbb{R}^d$, proving $c_{d,\alpha}(n)=\Theta\bigl(n^{\frac{d-1}{d+1}}\bigr)$ for every fixed $d\ge 2$ and $\alpha\ge 2$. The authors establish a matching lower bound via a probabilistic construction of ball-slice intersections that yields a positive transversal forming a convex independent set of size $\beta n^{\frac{d-1}{d+1}}$. For the upper bound, they extend Horton-type perturbations of grids to higher dimensions using oscillators and generalized convex cups/caps, then perform a detailed lattice-geometry analysis to show that any convex independent subset in the perturbed grid is at most $O\bigl(n^{\frac{d(d-1)}{d+1}}\bigr)$ in size, which, combined with density arguments, gives the required bound. Together, these results generalize Valtr’s planar construction and yield the exact growth rate in all dimensions, highlighting the role of high-dimensional perturbations and lattice packings in controlling convex configurations under diameter restrictions.
Abstract
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 $α\geq 2$.