等周问题的大偏差原理及其与非线性对数Sobolev不等式的等价性 (Large Deviations Principle for Isoperimetry and Its Equivalence to Nonlinear Log-Sobolev Inequalities)

from arxiv, We revised the light-tail condition to the weaker (better) one in (1.6); added Proposition 2 and the corresponding proof to better summarize the related inequalities; and also made other minor revisions

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.

翻译：本文研究了乘积黎曼流形 $M^{n}$ 上等周问题的大偏差原理（涉及指数小集合序列），其中流形配备乘积概率测度 $ν^{\otimes n}$，且 $M$ 为满足曲率-维度界 $\mathrm{CD}(0,\infty)$ 的黎曼流形。当概率测度 $ν$ 对平方距离具有有限矩生成函数时，我们建立了等周轮廓大偏差渐近行为的精确刻画，揭示了这些渐近等周不等式与非线性对数Sobolev不等式之间的严格等价关系。研究发现，两个相对熵典型集（或经验典型集）的乘积构成了等周问题的渐近最优解。本文证明过程综合运用了信息论、最优传输理论和几何测度论的工具。