Mathematics

I present some recent results, obtained in collaboration with Andrea Mondino and Daniele Semola, on Buser's and Cheeger's inequalities. We generalize to a possible non-smooth setting these two classical bounds involving the Cheeger's isoperimetric constant and the first eigenvalue of the Laplacian, in a way that Buser's inequality is now sharp in the class of ${\sf RCD}(K,\infty)$ spaces, $K >0$. We also discuss in detail the equality cases, showing that Buser's inequality is also rigid. Many of our results are new even in the smooth context of (weighted) Riemannian manifolds.