Sharp Cheeger-Buser Type Inequalities in RCD ( K , ∞ ) Spaces.

Abstract

The goal of the paper is to sharpen and generalise bounds involving Cheeger s isoperimetric constant h and the first eigenvalue λ 1 of the Laplacian. A celebrated lower bound of λ 1 in terms of h, λ 1 ≥ h 2 / 4 , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on λ 1 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 K ∈ R (the inequality is sharp for K > 0 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 RCD ( K , ∞ ) spaces.