Sharp Cheeger-buser type inequalities in \(\mathsf{RCD}(K,\infty)\) spaces (Q2659504)

This well written paper treats the improvement and variants of the so-called Cheeger and Buser inequalities concerning the lower and upper bounds on the first eigenvalue of the Riemannian Laplacian in terms of the Cheeger's isoperimetric constant. To show these improvements is the main objective of the article and is achieved in two directions. First of all, the setting the authors take up is on \textit{metric measure spaces} instead of Riemannian manifolds as is classically done in the context of Cheeger's and Buser's inequalities. The authors present the self contained proof of Cheeger's inequality on any complete metric measure space in the appendix of the paper. The main result of the paper demonstrates the Buser inequality with the dimension independent and improved, in comparison with the already known inequalities, constant. In condition, this result is shown to hold on certain metric measure spaces, called in the paper RCD spaces, instead of only Riemannian manifolds.