Quantitative estimates for the Bakry-Ledoux isoperimetric inequality

Content Provider	Semantic Scholar
Author	Mai, Cong Hung Ohta, Shin-Ichi
Copyright Year	2019
Abstract	We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry-Ledoux's Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag's needle decomposition (also called localization), and is inspired by the recent work of Cavalletti, Maggi and Mondino on compact spaces. At a key step, we obtain a reverse Poincare inequality for the guiding function, which is of independent interest.
File Format	PDF HTM / HTML
Alternate Webpage(s)	http://www4.math.sci.osaka-u.ac.jp/~sohta/papers/Qisop.pdf
Alternate Webpage(s)	https://arxiv.org/pdf/1910.13686v1.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article