No. A transportation cost inequality implies a strong concentration property (subgaussian concentration for Lipschitz functions) which most distributions don't satisfy. For example, an exponential distribution on $\mathbb{R}$ doesn't satisfy a transportation cost inequality.

Let's try again.

The theorem of Otto and Villani implies that every distribution $\nu$ which is log-concave in the sense you define satisfies a transportation-cost inequality.

There are many distributions $\mu$ with finite moments and density supported everywhere which do not satisfy a transportation-cost inequality. Since a TCI in particular implies subgaussian concentration, any distribution $\mu$ which has larger than Gaussian tails will fail to satisfy a TCI. One such distribution $\mu$ is the exponential distribution. But by the theorem of Otto and Villani, there is no such distribution $\mu$ which is also log-concave in the sense you define.

Whichever question you mean to ask, the answer is in the above two paragraphs.

No. A transportation cost inequality implies a strong concentration property (subgaussian concentration for Lipschitz functions) which most distributions, even log-concave distributions, don't satisfy. For example, an exponential distribution on $\mathbb{R}$ doesn't satisfy a transportation cost inequality.

No. A transportation cost inequality implies a strong concentration property (subgaussian concentration for Lipschitz functions) which most distributions don't satisfy. For example, an exponential distribution on $\mathbb{R}$ doesn't satisfy a transportation cost inequality.