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Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities
Abstract
Let Iinf(a):=limn→∞In(a)=infn≥1In(a) be the asymptotic isoperimetric profile for product of a weighted Riemannian manifold satisfying CD(0,∞) and, more broadly, for product of a nonsmooth subspace with density that can be approximated by a sequence of densities satisfying CD(0,∞). We establish the following identity: lima↓0Iinf(a)/a√(2ln(1/a))=√(KLS), where KLS is the optimal log-Sobolev constant. The same argument also yields the large-deviation and moderate-deviation asymptotics for isoperimetry.
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BibTeX
@misc{yu2026sharp,
title = {{Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities}},
author = {Lei Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23537},
eprint = {2609.23537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23537},
}
RIS
TY - UNPB
TI - Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities
AU - Lei Yu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23537
UR - https://arxiv.org/abs/2609.23537
AB - Let I_inf(a):=lim_n→∞I_n(a)=inf_n≥1I_n(a) be the asymptotic isoperimetric profile for product of a weighted Riemannian manifold satisfying CD(0,∞) and, more broadly, for product of a nonsmooth subspace with density that can be approximated by a sequence of densities satisfying CD(0,∞). We establish the following identity: lim_a↓0I_inf(a)/a√(2ln(1/a))=√(K_LS), where K_LS is the optimal log-Sobolev constant. The same argument also yields the large-deviation and moderate-deviation asymptotics for isoperimetry.
ER -