Quantitative estimates for the Bakry-Ledoux isoperimetric inequality. II

TL;DR

This paper provides quantitative estimates for the Bakry-Ledoux isoperimetric inequality on weighted Riemannian manifolds with Ricci curvature at least 1, demonstrating the measure's proximity to Gaussian distribution through various mathematical estimates.

Contribution

It introduces new $L^1$, $L^p$, and $W_2$ estimates that quantify how the measure related to the isoperimetric inequality approximates Gaussian behavior.

Findings

The push-forward measure is close to Gaussian in $L^1$ sense.

Established $L^p$ estimates in the 1-dimensional case.

Derived $W_2$-estimates showing measure proximity to Gaussian.

Abstract

Concerning quantitative isoperimetry for a weighted Riemannian manifold satisfying $\mathrm{Ric}_{\infty} \ge 1$, we give an $L^1$-estimate exhibiting that the push-forward of the reference measure by the guiding function (arising from the needle decomposition) is close to the Gaussian measure. We also show $L^p$- and $W_2$-estimates in the $1$-dimensional case.