Isoperimetric Bounds on Convex Manifolds

Abstract

We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lov\'asz-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require in addition an upper bound on the sectional curvature of the space, which permits us to use comparison tools in Cartan-Alexandrov-Toponogov (or CAT) spaces. Along the way, we also quantitatively improve our previous result that weak concentration assumptions imply a Cheeger-type isoperimetric bound, to a sharp bound with respect to all parameters.