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arXiv · 2609.23537

Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities

Lei Yu
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Abstract

Let $I_{\inf}(a):=\lim_{n\to\infty}I_{n}(a)=\inf_{n\ge1}I_{n}(a)$ be the asymptotic isoperimetric profile for product of a weighted Riemannian manifold satisfying $\mathrm{CD}(0,\infty)$ and, more broadly, for product of a nonsmooth subspace with density that can be approximated by a sequence of densities satisfying $\mathrm{CD}(0,\infty)$. We establish the following identity: \[ \lim_{a\downarrow0}\frac{I_{\inf}(a)}{a\sqrt{2\ln(1/a)}}=\sqrt{K_{\mathrm{LS}}}, \] where $K_{\mathrm{LS}}$ is the optimal log-Sobolev constant. The same argument also yields the large-deviation and moderate-deviation asymptotics for isoperimetry.

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Lei Yu. 2026-09-20. Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities. https://arxiv.org/abs/2609.23537

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