Optimal Spectral Bounds for Antipodal Graphs

Abstract: Suppose $\left{x_1, \dots, x_n\right} \subset \mathbb{R}<sup>2$ 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 the ratio of the number of "neighbors" (pairs of points with distance $\leq \varepsilon$) to the number of "antipodes" (pairs of points with distance $\geq 1 - \varepsilon$) is $\gtrsim\varepsilon^{1/2 + o(1)}$, attaining the conjectured correct asymptotic within a polylog factor and improving the $\gtrsim\varepsilon<sup>{3/4+o(1)}$ bound of Steinerberger (2025).