The isoperimetric problem is one of the most classical problems in mathematics. It addresses the following natural problem:

given a space $X$, find the minimal amount of area needed to enclose a fixed volume $v$.

If the space $X$ has a simple structure or many symmetries, the problem can be completely solved and the "optimal shapes" can be explicitly described (e.g. Euclidean space and the sphere). In the general case, however, one cannot hope to obtain a complete solution to the problem and a comparison result is already satisfactory. Probably, the most popular result in this direction is the Levy-Gromov isoperimetric inequality.

During the talk, we will show that a sharp isoperimetric inequality à la Levy-Gromov holds true in the class of essentially non-branching metric measure spaces $(X,\mathsf d,\mathfrak m)$ with $\mathfrak m(X)=1$ satisfying the so called Measure-Contraction property, with the latter being a condition that, in a synthetic way, encodes bounds on the Ricci curvature and on the dimension of the space. Measure theoretic rigidity is also obtained.

*This is a joint work with prof. Fabio Cavalletti.*