It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ satisfying the curvature condition 

$$
\operatorname{Hess}_x(V) + \operatorname{Ric}_x \succeq \rho I_n,\forall x \in X,
$$

the *transportation-cost inequality*
$$W(\nu,\mu) \le \sqrt{(2/\rho)H(\nu\|\mu)}
$$
holds for all other distributions $\nu$ on $X$. Here $W$ is the Wasserstein distance induced by the geodesic metric on $X$.

Question
========
- *(A)* What are the most general conditions under which the above transportation-cost inequality holds.

- *(B)* Can the curvature condition be relaxed to "piecewise" version. That is, what if we instead assume a convex mixture $d\mu = \sum_{i=1}^k \pi_i d\mu_i$ where each $d\mu_i$ is log-concave and satisfies the curvature condition on some piece $X_i$ of $X$ ?

- *(B')* Are there concentration inequalities for mixtures of Gaussians ?