Sharp Antipode-Neighbor Bounds in Every Dimension

Abstract

Let $P\subset {\mathbb{R}}^d$ be finite with $\mathrm{diam}(P)\le 1$. For $0<\epsilon <1$, let ${\mathcal{N}}_{\epsilon }(P)$ denote the number of ordered pairs at distance at most $\epsilon$, including the diagonal, and let ${\mathcal{A}}_{\epsilon }(P)$ denote the number of ordered pairs at distance at least $1-\epsilon$. We prove the sharp bound \[ \mathcal N_\varepsilon(P)\gtrsim_d \varepsilon^{(d-1)/2}\mathcal A_\varepsilon(P) \] conjectured by Steinerberger (2025). More generally, the same inequality holds for finite positive Borel measures. The main input is a uniform $L^2$ estimate, at cap thickness comparable to lattice spacing, for a supporting-direction incidence operator. In gnomonic coordinates, this operator is controlled by an incidence estimate for small Fenchel--Young gaps.