Large Deviations Principle for Isoperimetry and Its Equivalence to Nonlinear Log-Sobolev Inequalities
Lei Yu
TL;DR
This work characterizes the large deviations behavior of the isoperimetric profile on high-dimensional product manifolds with CD(0,∞) density bounds, tying the asymptotics to nonlinear log-Sobolev inequalities. By weaving together nonlinear‑entropy, transport-entropy, and geometric measure theory techniques, it proves an exact equivalence: the isoperimetric large deviations exponent Λ(α) equals the square root of a lower convex envelope breveΘ(α). It further develops a comprehensive web of constants (K_CD,K_LS,K_IS,K_T, etc.) and demonstrates dimension-free and infinite-dimensional correspondences, including a Φ-Sobolev generalization and interpretations via typical sets. The results illuminate deep connections between isoperimetry, nonlinear Sobolev inequalities, and transport costs, with implications for concentration phenomena in high dimensions.
Abstract
We investigate the large deviations principle (which concerns sequences of exponentially small sets) for the isoperimetric problem on product Riemannian manifolds $M^{n}$ equipped with product probability measures $ν^{\otimes n}$, where $M$ is a Riemannian manifold satisfying curvature-dimension bound $\mathrm{CD}(0,\infty)$. When the probability measure $ν$ admits a finite moment generating function for squared distance, we establish an exact characterization of the large deviations asymptotics for the isoperimetric profile, which shows a precise equivalence between these asymptotic isoperimetric inequalities and nonlinear log-Sobolev inequalities. It is observed that the product of two relative entropy typical sets (or empirically typical sets) forms an asymptotically optimal solution to the isoperimetric problem. The proofs in this paper integrate tools from information theory, optimal transport, and geometric measure theory.