Spectral Bounds for Antipodal Graphs

Abstract

Suppose $\left\{x_1,\dots ,x_n\right\}\subset {\mathbb{R}}^2$ is a set of $n$ points in the plane with diameter $\le 1$, meaning $|x_i-x_j|\le 1$ for all $1\le i,j\le n$. We show that the ratio of the number of ``neighbors'' (ordered pairs of points with distance $\le \epsilon$) to the number of ``antipodes'' (ordered pairs of points with distance $\ge 1-\epsilon$) is $\gtrsim {\epsilon }^{1/2+o(1)}$, attaining the conjectured correct asymptotic within a polylog factor and improving the $\gtrsim {\epsilon }^{3/4+o(1)}$ bound of Steinerberger (2025). In dimensions $d\ge 3$ we prove a similar result with exponent $3(d-1)/4$.