# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.331969

Acta Mathematica, 196, (1), 65-131, 2004.10
We introduce and analyze lower ($Ricci$) curvature bounds $\underline{{Curv}} {\left( {M,d,m} \right)}$ ⩾$K$for metric measure spaces ${\left( {M,d,m} \right)}$ . Our definition is based on convexity properties of the relative entropy $Ent{\left( { \cdot \left| m \right.} \right)}$ regarded as a function on the$L$_{2}-Wasserstein space of probability measures on the metric space ${\left( {M,d} \right)}$ . Among others, we show that $\underline{{Curv}} {\left( {M,d,m} \right)}$ ⩾$K$implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $\underline{{Curv}} {\left( {M,d,m} \right)}$ ⩾$K$if and only if $Ric_{M} {\left( {\xi ,\xi } \right)}$ ⩾$K$ ${\left| \xi \right|}^{2}$ for all $\xi \in TM$ .
@inproceedings{karl-theodor2004on,
title={On the geometry of metric measure spaces},
author={Karl-Theodor Sturm},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203348822929678},
booktitle={Acta Mathematica},
volume={196},
number={1},
pages={65-131},
year={2004},
}

Karl-Theodor Sturm. On the geometry of metric measure spaces. 2004. Vol. 196. In Acta Mathematica. pp.65-131. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203348822929678.