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 $ν^{\otimes n}$, where $(M,ν)$ is a weighted Riemannian manifold with nonnegative Bakry--Émery--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.

翻译：等周问题是几何测度论中的经典课题，但关于最优解——甚至渐近最优解——的表征这一关键问题在很大程度上仍未得到解决。本文研究了乘积黎曼流形 $M^{n}$ 上的等周问题的大偏差渐近性，其中 $M^{n}$ 赋予乘积概率测度 $ν^{\otimes n}$，而 $(M,ν)$ 是一个具有非负 Bakry–Émery–Ricci 曲率的加权黎曼流形。我们建立了等周剖面大偏差渐近性的精确表征，揭示了这一渐近等周不等式与非线性对数索博列夫不等式之间的精确等价关系。研究发现，条件典型集——信息论中的一个基本概念——构成了等周问题的一个渐近最优解。这类子集进一步给出了中心极限区域中等周剖面的一个上界。尽管我们的结果针对乘积空间陈述，但它们也暗示了非乘积空间的某些等周不等式，例如，它们可用于恢复 Ledoux 和 Bobkov 为任意非乘积空间建立的较弱等价性，或用于建立乘积或非乘积空间中等周、集中和输运不等式最优常数之间的定量关系。我们的结果从非线性对数索博列夫不等式的角度，为等周极小元在具有不同几何结构的空间中表现出根本性差异提供了严格的理论依据。我们的证明思路是一个整合了信息论、最优输运和几何测度论工具的新框架。