*(English)*Zbl 1178.53038

The main results of this very interesting paper, showing that there is a good notion of a compact measured length space having “$N$-Ricci curvature bounded below by $K\in \mathbb{R}$” for each $N\in [1,+\infty ]$ – via optimal transport in the setting of length spaces – which is accompanied with geometric and analytic consequences, have been presented in a previous paper of the first author [“Optimal transport and Ricci curvature for metric-measure spaces”, see our review of this paper, that appeared in Surveys in Differential Geometry 11, 229–257 (2007; Zbl 1155.53026)].

The authors provide here complete proofs of these results. In the case of Riemannian manifolds their definitions are equivalent to classical ones. There are many other aspects of the paper, and in particular, six appendices that contain technical and auxiliary results, explanations how to extend the results of the paper from the setting of compact measured length spaces to the setting of complete pointed locally compact measured length spaces, and some bibliographic notes on optimal transport and displacement convexity, which illustrate the authors theory in a convincing way.

##### MSC:

53C23 | Global geometric and topological methods; differential geometric analysis on metric spaces |

53C21 | Methods of Riemannian geometry, including PDE methods; curvature restrictions (global) |

60B05 | Probability measures on topological spaces |

60B10 | Convergence of probability measures |

58C35 | Integration on manifolds; measures on manifolds |