The paper by John Lott and Cédric Villani introduces a notion of Ricci curvature for measured length spaces, extending the classical definition from smooth Riemannian manifolds to more general metric spaces. The authors define nonnegative $N$-Ricci curvature for a compact measured length space $(X, d, \nu)$ if for all $\mu_0, \mu_1 \in P_2(X)$ with $\text{supp}(\mu_0), \text{supp}(\mu_1) \subset \text{supp}(\nu)$, there exists a Wasserstein geodesic $\{\mu_t\}_{t \in [0, 1]}$ such that for all $U \in \mathcal{DC}_N$ and $t \in [0, 1]$, $$ U_\nu(\mu_t) \leq t U_\nu(\mu_1) + (1-t) U_\nu(\mu_0), $$ where $U_\nu(\mu) = \int_X U(\rho(x)) \, d\nu(x) + U'(\infty) \, \mu_s(X)$ and $\mu = \rho \nu + \mu_s$. They also define $\infty$-Ricci curvature bounded below by $K$ if for all $\mu_0, \mu_1 \in P_2(X)$ with $\text{supp}(\mu_0), \text{supp}(\mu_1) \subset \text{supp}(\nu)$, $$ U_\nu(\mu_t) \leq t U_\nu(\mu_1) + (1-t) U_\nu(\mu_0) - \frac{1}{2} \lambda(U) t(1-t) W_2(\mu_0, \mu_1)^2, $$ where $\lambda(U)$ is a function depending on $U$. The paper shows that these definitions are preserved under measured Gromov–Hausdorff limits and provides geometric and analytic consequences, such as log Sobolev inequalities and Poincaré inequalities. The authors also establish connections to classical notions of Ricci curvature for smooth Riemannian manifolds, proving that their definitions are equivalent under certain conditions.The paper by John Lott and Cédric Villani introduces a notion of Ricci curvature for measured length spaces, extending the classical definition from smooth Riemannian manifolds to more general metric spaces. The authors define nonnegative $N$-Ricci curvature for a compact measured length space $(X, d, \nu)$ if for all $\mu_0, \mu_1 \in P_2(X)$ with $\text{supp}(\mu_0), \text{supp}(\mu_1) \subset \text{supp}(\nu)$, there exists a Wasserstein geodesic $\{\mu_t\}_{t \in [0, 1]}$ such that for all $U \in \mathcal{DC}_N$ and $t \in [0, 1]$, $$ U_\nu(\mu_t) \leq t U_\nu(\mu_1) + (1-t) U_\nu(\mu_0), $$ where $U_\nu(\mu) = \int_X U(\rho(x)) \, d\nu(x) + U'(\infty) \, \mu_s(X)$ and $\mu = \rho \nu + \mu_s$. They also define $\infty$-Ricci curvature bounded below by $K$ if for all $\mu_0, \mu_1 \in P_2(X)$ with $\text{supp}(\mu_0), \text{supp}(\mu_1) \subset \text{supp}(\nu)$, $$ U_\nu(\mu_t) \leq t U_\nu(\mu_1) + (1-t) U_\nu(\mu_0) - \frac{1}{2} \lambda(U) t(1-t) W_2(\mu_0, \mu_1)^2, $$ where $\lambda(U)$ is a function depending on $U$. The paper shows that these definitions are preserved under measured Gromov–Hausdorff limits and provides geometric and analytic consequences, such as log Sobolev inequalities and Poincaré inequalities. The authors also establish connections to classical notions of Ricci curvature for smooth Riemannian manifolds, proving that their definitions are equivalent under certain conditions.