  We investigate separation properties of $N$-point configurations that
minimize 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 the
best possible. The same conclusions hold for the points that are a fixed
positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional
set. These estimates extend a result of Dahlberg for certain smooth
$(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we
obtain the same separation results for `greedy' $s$-energy points. We deduce
our results from an upper regularity property of the $s$-equilibrium measure
(i.e., the measure that solves the continuous minimal Riesz $s$-energy
problem), and we show that this property holds under a local smoothness
assumption on the set $A$.
