In this paper, the authors apply previous estimates they obtained in [Comment. Math. Helv. 96, No. 4, 693–739 (2021; Zbl 1486.53049)] to some quantitative version of the isoperimetric inequality of D. Bakry and M. Ledoux [Invent. Math. 123, No. 2, 259–281 (1996; Zbl 0855.58011)]. More precisely, let \((M,g)\) be a weighted Riemannian manifold equipped with the measure \(m=e^{-\psi} \mathrm{vol}_g\) where \(\psi\in C^\infty(M)\). Assuming \(\mathrm{Ric}_\infty \geq 1\), the Bakry-Émery isoperimetric inequality holds true. The aim of the paper is to get an \(L^1\)-bound between \(\gamma=\frac{1}{\sqrt{2\pi}}\,e^{-x^2/2} dx\) and the push-forward measure \(u_*m\) of \(m\) by the guiding function \(u\) arising in the needle decomposition. They prove that \[ \|\rho e^{\psi_g}-1\|_{L^1(\gamma)} \leq C(\theta,\varepsilon) \delta^{(1-\varepsilon)/(9-3\varepsilon)}, \] where \(u_*m=\rho dx\) and \(\gamma=e^{-\psi_\varepsilon} dx\).
They also study in detail the one-dimensional case and prove, in particular, an \(L^p\)-bound with the sharp order \(\delta^{1/p}\).
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