Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds (Q1650834)

For \(K>0\) it is known by the Levy-Gromov isoperimetric inequality that for \(N\)-dimensional Riemannian manifolds of Ricci curvature \(\operatorname{Ric}\geq K\) the isoperimetric profile function is bounded below by that of an \(N\)-dimensional round sphere. For arbitrary \(K\in{\mathbb R}\) the analogous lower bound \({\mathcal I}_{K,N,D}\) has recently be found by \textit{E. Milman} [J. Eur. Math. Soc. (JEMS) 17, No. 5, 1041--1078 (2015; Zbl 1321.53043)]. The paper under review considers more general metric measure spaces \((X,d,m)\) with \(m(X)=1\) and diameter \(D\) possibly infinite, which satisfy the essentially nonbranching property and \(CD_{\mathrm{loc}}(K,N)\). It proves a generalization of the Levy-Gromov inequality in this context: for every Borel set \(E\subset X\) the isoperimetric inequality \(P(E)\geq {\mathcal I}_{K,N,D}(v)\) for \(v=m(E)\) holds, where the perimeter \(P(E)\) is defined as in [\textit{L. Ambrosio}, Adv. Math. 159, No. 1, 51--67 (2001; Zbl 1002.28004)]. Moreover the authors prove some rigidity results, saying that equality in the generalized Levy-Gromov inequality implies that \((X,d,m)\) is a spherical suspension, and that almost equality implies almost maximal diameter and measured-Gromov-Hausdorff-closeness to a spherical suspension.