The following is one example of a transportation-information inequality, which show connections between optimal transport theory and information theory:
$$W_1(\nu,\mu) \leq \left[ 2\sigma^2 D_{KL}(\nu\Vert\mu) \right]^\frac{1}{2} $$
$W_1$ is the Wasserstein distance found in optimal transport theory, and $D_{KL}$ is the Kullback-Leibler (KL) divergence found in information theory. (Source)
What other transportation-information inequalities are out there?
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From this answer regarding books on Concentration of Measure, the following textbook gave good coverage of many transportation-information linkages, having full chapters on both the entropy approach and the transportation approach.
although I could see it to be inaccessible to those new to the field since alot of the theoretical derivations of the inequalities stop short of providing the completed forms of information theoretic measures we are more familiar with. There is also a lack of actual applications, given that concentration of measure is a construct of independent variables, so other recommended textbooks on applied transportation-information would be helpful.
There are several other Transportation Inequalities. I recommend looking this paper: Transport Inequalities. A Survey.
The authors not only show different transport inequalities (e.g. Talagrand’s transport inequality for the Gaussian measure), but also prove them.