Starting with the work of Bavard-Pansu in the eighties it was understood that a lower bound on the Ricci curvature of a smooth Riemannian manifold leads to a sharp concavity inequality for its isoperimetric profile. In joint work with Antonelli, Pasqualetto, and Pozzetta we generalized such inequality to the case of $\mathrm{RCD}(K,N)$ spaces, for $N<\infty$. In this talk, I will review the problem and discuss some open questions that are left if either the Riemannian or the finite dimensionality assumption is dropped.