Large Deviations Principle for Isoperimetry and Its Equivalence to Nonlinear Log-Sobolev Inequalities

Abstract

The isoperimetric problem is a classic topic in geometric measure theory, yet critical questions regarding the characterization of optimal solutions -- even asymptotically optimal ones -- remain largely unresolved. In this paper, we investigate the large deviations asymptotics for the isoperimetric problem on the product Riemannian manifold $M^{n}$ endowed with the product probability measure $\nu^{\otimes n}$, where $(M,\nu)$ is a weighted Riemannian manifold with nonnegative Bakry--\'Emery--Ricci curvature. We establish an exact characterization of the large deviations asymptotics of the isoperimetric profile, which reveals a precise equivalence between this asymptotic isoperimetric inequality and the nonlinear log-Sobolev inequality. It is observed that conditional typical sets, a fundamental concept from information theory, form an asymptotically optimal solution to the isoperimetric problem. This class of subsets further yields an upper bound on the isoperimetric profile in the central limit regime. Although our results are stated for product spaces, they imply certain isoperimetric inequalities for non-product spaces, e.g., they can be used to recover the weaker equivalence established by Ledoux and Bobkov for arbitrary non-product spaces or used to establish quantitative relations among the optimal constants in isoperimetric, concentration and transport inequalities for product or non-product spaces. Our results provide a rigorous justification from the perspective of nonlinear log-Sobolev inequalities for why isoperimetric minimizers behave fundamentally differently across spaces with distinct geometric structures. Our proof idea is a new framework which integrates tools from information theory, optimal transport, and geometric measure theory.