  We establish quantitative properties of minimizers and stable sets for
nonlocal interaction functionals, including the $s$-fractional perimeter as a
particular case.
  On the one hand, we establish universal $BV$-estimates in every dimension
$n\ge 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has
finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal
result is new even in the case of $s$-perimeters and its local counterpart (for
classical stable minimal surfaces) was known only for simply connected
two-dimensional surfaces immersed in $\mathbb R^3$.
  On the other hand, we prove quantitative flatness estimates for minimizers
and stable sets in low dimensions $n=2,3$. More precisely, we show that a
stable set in $B_R$, with $R$ large, is very close in measure to being a half
space in $B_1$ ---with a quantitative estimate on the measure of the symmetric
difference. As a byproduct, we obtain new classification results for stable
sets in the whole plane.
