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 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 $3(d - 1)/4$.
From: Samuel Korsky [view email]
[v1]
Wed, 11 Mar 2026 02:06:10 UTC (5 KB)
[v2]
Mon, 18 May 2026 06:23:40 UTC (6 KB)
[v3]
Fri, 3 Jul 2026 05:44:55 UTC (6 KB)
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