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We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with$\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded frombelow by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then theclassic L\'evy-Gromov isoperimetric inequality (together with the recentsharpening counterparts proved in the smooth setting by E. Milman for any $K\in\mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetricprofile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by theisoperimetric profile of the model space. Moreover, if equality is attained forsome volume $v \in (0,1)$ and $K$ is strictly positive, then the space must bea spherical suspension and in this case we completely classify theisoperimetric regions. Finally we also establish the almost rigidity: if theequality is almost attained for some volume $v \in (0,1)$ and $K$ is strictlypositive, then the space must be mGH close to a spherical suspension.  To our knowledge this is the first result about isoperimetric comparison fornon smooth metric measure spaces satisfying Ricci curvature lower bounds.Examples of spaces fitting our assumptions include measured Gromov-Hausdorfflimits of Riemannian manifolds satisfying Ricci curvature lower bounds andAlexandrov spaces with curvature bounded from below; the result seems new evenin these celebrated classes of spaces.

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds