  This paper studies sharp isoperimetric comparison theorems and sharp
dimensional concavity properties of the isoperimetric profile for non smooth
spaces with lower Ricci curvature bounds, the so-called $N$-dimensional ${\rm
RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. The absence of most of the
classical tools of Geometric Measure Theory and the possible non existence of
isoperimetric regions on non compact spaces are handled via an original
argument to estimate first and second variation of the area for isoperimetric
sets, avoiding any regularity theory, in combination with an asymptotic mass
decomposition result of perimeter-minimizing sequences. Most of our statements
are new even for smooth, non compact manifolds with lower Ricci curvature
bounds and for Alexandrov spaces with lower sectional curvature bounds. They
generalize several results known for compact manifolds, non compact manifolds
with uniformly bounded geometry at infinity, and Euclidean convex bodies.
