Synthetic Ricci Curvature Bounds on Metric Measure Spaces and the Levy-Gromov Isoperimetric Inequality

This monograph explores the extension of the classical Lévy-Gromov isoperimetric inequality to the non-smooth setting of metric measure spaces satisfying synthetic Ricci curvature bounds. We analyze the Lott-Sturm-Villani Curvature-Dimension condition CD(K,N) and its Riemannian refinement RCD*(K,N), which characterizes spaces with Ricci curvature bounded below by K and dimension bounded above by N via optimal transport and entropic convexity. The core of the work focuses on the localization paradigm developed by Cavalletti and Mondino, utilizing L1-optimal transport to decompose the ambient space into one-dimensional "needles" (geodesics with log-concave measures). We demonstrate how this disintegration technique reduces the high-dimensional isoperimetric problem to a family of one-dimensional inequalities, thereby establishing sharp isoperimetric bounds and rigidity results for RCD*(K,N) spaces that mirror the smooth Riemannian case. The structural stability of these bounds under measured Gromov-Hausdorff convergence is also discussed.