Many antipodes implies many neighbors

Abstract

Suppose $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$ is a set of $n$ points in the plane with diameter $\leq 1$, meaning $\|x_i - x_j\| \leq 1$ for all $1 \leq i,j \leq n$. We show that if there are many `antipodes', these are pairs of points of with distance $\geq 1-\varepsilon$, then there are many neighbors, these are pairs of points that are distance $\leq \varepsilon$. More precisely, we prove that for some universal $c>0$, $$ \# \left\{(i,j): \|x_i - x_j\| \leq \varepsilon\right\} \geq \frac{c \cdot \varepsilon^{3/4}}{\left( \log \varepsilon^{-1} \right)^{1/4}}\cdot \# \left\{(i,j): \|x_i - x_j\| \geq 1- \varepsilon\right\}.$$ The inequality is very easy too prove with factor $\varepsilon^2$ and easy with $\varepsilon$. The optimal rate might be $\varepsilon^{1/2}$ which is attained by several examples.