Spectral Bounds for Antipodal Graphs

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