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

$$ \operatorname{Hess}_x(V) + \operatorname{Ric}_x \succeq (1/c) I_n,\forall x \in X, $$

the *transportation-cost inequality*
$$W_2(\nu,\mu) \le \sqrt{2cH(\nu\|\mu)}
$$
holds for all other distributions $\nu$ on $X$ absolutely continuous w.r.t $\mu$. Here $W_2$ is the Wasserstein-2 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 ?

# Partial answer

- Log-concave distributions satisfying the Bakry-Eméry $\operatorname{CD}(n,\infty)$ curvature condition on manifolds
- Distributions with densities on compact homogeneous Riemannian manifolds. See this paper of Rothaus.
- Distributions which can be realized as pushforwards of distributions with some $\text{T}_2(c)$, under Lipschitz maps. If $\mu$ has $\text{T}_2(c)$ property and $\varphi: X \rightarrow Y$ is $L$-Lipschitz, then $\varphi_\#\mu$ has $\text{T}_2(L^2c)$.
- Finite tensor product $\mu_1 \otimes \mu_2 \otimes \ldots \otimes \mu_k$ of distributions having $\text{T}_2(c)$ also has $\text{T}_2(c)$.