Define the Wasserstein-1 metric (or the Earth mover's distance) between two positive measures $\mu\_1$, $\mu\_2$ by


$$
W(\mu\_1, \mu\_2)
=
\inf\_{\gamma \in \Gamma (\mu\_1, \mu\_2)} \int \|x\_1 - x\_2\| \, \mathrm{d} \gamma (x\_1, x\_2)
$$
where $\Gamma(\mu\_1,\mu\_2)$ denotes the collection of all measures on with marginals $\mu\_1$ and $\mu\_2$.


Is the Wasserstein-1 metric translation invariant? E.g. if $\nu$ is some positive measure, does it hold that
$$
 W(\mu\_1, \mu\_2) = W(\mu\_1 + \nu, \mu\_2 + \nu)
$$



Is the Wasserstein-1 metric translation invariant?
Yes, by Kantorovich--Rubinstein duality $W(\mu\_1,\mu\_2)=\sup\_{f\,\text{is 1-Lip}} \int f d(\mu\_1-\mu\_2)$.


