Sharp Antipode-Neighbor Bounds in Every Dimension

Let $P\subset\mathbb R^d$ be finite with $\operatorname{diam}(P)\le1$. For $0<\varepsilon<1$, let $\mathcal N_\varepsilon(P)$ denote the number of ordered pairs at distance at most $\varepsilon$, including the diagonal, and let $\mathcal A_\varepsilon(P)$ denote the number of ordered pairs at distan