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

TL;DR

This paper investigates the size of convex independent subsets in finite point sets in $ ext{R}^d$ with bounded diameter ratios, establishing asymptotic bounds that depend on the number of points and the dimension.

Contribution

It determines the asymptotic behavior of the maximum convex independent subset size in restricted point sets in $ ext{R}^d$, extending understanding of geometric configurations under diameter constraints.

Findings

Asymptotic bounds for $c_{d, ext{alpha}}(n)$ are established.

The bounds are proportional to $n^{(d-1)/(d+1)}$ for $ ext{alpha} \

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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 $c_{d, \alpha}(n)$ 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 $\alpha\geq 2$.