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

We investigate the large deviations principle (which concerns sequences of exponentially small sets) for the isoperimetric problem on product Riemannian manifolds Mⁿ equipped with product probability measures ν⊗ n, where M is a Riemannian manifold satisfying curvature-dimension bound CD(0,∞). When the probability measure $ν$ satisfies a specific light-tail condition, 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 their one-sided versions (or the product of two empirically typical sets) forms an asymptotically optimal solution to the isoperimetric problem. The proofs in this paper rely on tools from information theory, optimal transport, and geometric measure theory.