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

}, keywords = {Isoperimetric inequality, Measure-Contraction property, Optimal transport, Ricci curvature}, isbn = {0022-1236}, url = {https://www.sciencedirect.com/science/article/pii/S0022123619302289}, author = {Fabio Cavalletti and Flavia Santarcangelo} }