Isoperimetric inequality under Measure-Contraction property

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,∞) , understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.