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We investigate separation properties of $N$-point configurations thatminimize discrete Riesz $s$-energy on a compact set $A\subset \mathbb{R}^p$.When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $s\in[p-2, p-1)$, we prove that the order of separation (as $N\to \infty$) is thebest possible. The same conclusions hold for the points that are a fixedpositive distance from the boundary of $A$ whenever $A$ is any $p$-dimensionalset. These estimates extend a result of Dahlberg for certain smooth$(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, weobtain the same separation results for `greedy' $s$-energy points. We deduceour results from an upper regularity property of the $s$-equilibrium measure(i.e., the measure that solves the continuous minimal Riesz $s$-energyproblem), and we show that this property holds under a local smoothnessassumption on the set $A$.

Local properties of Riesz minimal energy configurations and equilibrium measures