Sharp Cheeger–Buser Type Inequalities in [Formula: see text] Spaces

The goal of the paper is to sharpen and generalise bounds involving Cheeger’s isoperimetric constant h and the first eigenvalue [Formula: see text] of the Laplacian. A celebrated lower bound of [Formula: see text] in terms of h, [Formula: see text] , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on [Formula: see text] in terms of h was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is twofold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry–Émery weighted) Ricci curvature bounded below by [Formula: see text] (the inequality is sharp for [Formula: see text] as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called [Formula: see text] spaces.