Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound

Abstract

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry--Émery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension \emph{independent} bounds. As a corollary, we can recover and extend all previously known (dimension dependent) results by generalizing an isoperimetric inequality of Bobkov, and provide a new proof that under natural convexity assumptions, arbitrarily weak concentration implies a dimension independent linear isoperimetric inequality. Further applications will be described in a subsequent work. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifold-with-density setting by Morgan.

28 pages; to appear in Duke Math. J. - shortened exposition and addressed referees' useful comments

References in corpus (3)

• On the role of Convexity in Functional and Isoperimetric Inequalities
• Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds
• Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration

Cited by in corpus (14)

• Pointwise Symmetrization Inequalities for Sobolev functions and applications
• Transport Inequalities. A Survey
• Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems
• A new characterization of Talagrand's transport-entropy inequalities and applications
• Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds
• Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities
• Moment measures and stability for Gaussian inequalities
• Concentration of $1$-Lipschitz functions on manifolds with boundary with Dirichlet boundary condition
• On the equivalence of modes of convergence for log-concave measures
• The Legendre Transform in Non-additive Thermodynamics and Complexity
• Multiple sets exponential concentration and higher order eigenvalues
• Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration
• Concentration of measures supported on the cube
• The KLS Isoperimetric Conjecture for Generalized Orlicz Balls