Spectral Bounds for Antipodal Graphs

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 the ratio of the number of ``neighbors'' (ordered pairs of points with distance $\leq \varepsilon$) to the number of ``antipodes'' (ordered 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 $\gtrsim\varepsilon^{3/4+o(1)}$ bound of Steinerberger (2025). In dimensions $d\ge3$ we prove a similar result with exponent $\min\left\{d-3/2,\ 3(d - 1)/4\right\}$.