Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities

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 den