  In this paper we study regularity and topological properties of volume
constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the
reference measure is the Hausdorff measure. A quasi-perimeter is a functional
given by the sum of the usual perimeter and of a suitable continuous term. In
particular, isoperimetric sets are a particular case of our study.
  We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathcal{H}^N)$,
with $K\in\mathbb R$, $N\geq 2$, and a uniform bound from below on the volume
of unit balls, volume constrained minimizers of quasi-perimeters are open
bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with
the essential boundary.
  The proof is based on a new Deformation Lemma for sets of finite perimeter in
${\sf RCD}(K,N)$ spaces $({\rm X},{\sf d},\mathfrak m)$ and on the study of
interior and exterior points of volume constrained minimizers of
quasi-perimeters.
  The theory applies to volume constrained minimizers in smooth Riemannian
manifolds, possibly with boundary, providing a general regularity result for
such minimizers in the smooth setting.
