In this talk, I shall discuss two results that show how the isoperimetric structure of a space is connected to its geometry.

First, I will show a sharp concavity property of the isoperimetric profile of manifolds with Ricci lower bounds. Although the statement is set in the smooth context, its proof crucially relies on tools from non-smooth geometry that have been developed in recent years, along with a new understanding of mean curvature in metric measure spaces. I will explain how this concavity result can be used to address the existence, non-existence, and uniqueness of isoperimetric regions in spaces with lower curvature bounds.

Next, I will present a sharp and rigid generalization of the Bishop-Gromov volume comparison theorem. The proof of this result builds on a concavity property of an unequally weighted isoperimetric profile on the manifold, similar to the one mentioned above. I will then discuss how this volume estimate has been recently used by L. Mazet, following contributions by O. Chodosh, C. Li, P. Minter, and D. Stryker, to settle a well-known open problem: the stable Bernstein problem in R^n, with n<=6.

If time permits, I will touch on open questions and perspectives regarding the results presented. The results I will discuss are based on joint works with E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, M. Pozzetta, D. Semola, I. Y. Violo, and K. Xu.