Isoperimetric inequality under Measure Contraction property
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.
Journal of Functional Analysis