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

Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds