The Abresch-Gromoll inequality in a non-smooth setting.

*(English)*Zbl 1280.53038The author gets an inequality for metric measure spaces in the class of infinitesimally Hilbertian spaces, which are mm-spaces with Ricci curvature bounded from below and dimension bounded from above. This work is a non-trivial extension of a result of U. Abresch and D. Gromoll [J. Am. Math. Soc. 3, No. 2, 355–374 (1990; Zbl 0704.53032)]. The authors develop a method based on a deep analysis of Abresch and Gromoll’s proof, and check step by step that the key arguments that they identified are also true in the framework considered.

The methods employed makes this papers very interesting for beginners in mm-spaces, because of the exposition of standard tools.

The methods employed makes this papers very interesting for beginners in mm-spaces, because of the exposition of standard tools.

Reviewer: Jean-Pierre Magnot (Clermont-Ferrand)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |

53C70 | Direct methods (\(G\)-spaces of Busemann, etc.) |

46E27 | Spaces of measures |

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