Zichao Dong, Convex polytopes in non-elongated point sets in $\mathbb{R}^d$

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For any finite point set $P \subset \mathbb{R}^d$, we 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 $\text{diam}(P) < \alpha\sqrt{n}$ (informally speaking, `non-elongated'), contains a

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