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Consider the Wasserstein metric of order one $W_1$ (aka the Earth Movers Distance). I would like to know whether it is possible to link $W_1$ and relative entropy and what this would mean intuitively. I can't find it anymore, but if I am not mistaken the following holds true for some constant $C$ $$ W_1(\mu, \nu)\le \sqrt{C\cdot \text{KL}(\nu ||\mu)}, $$

where KL is the Kullback–Leibler divergence. My first question would be: Is the above-mentioned inequality true? Secondly, how should one interpret this estimation?

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