ZichaoDong

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 …