Isoperimetric inequality under Measure-Contraction property

Abstract: We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality `a la L\'evy-Gromov holds true. Measure theoretic rigidity is also obtained.