# Is the Wasserstein-1 metric translation invariant?

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 $$W(\mu_1, \mu_2) = W(\mu_1 + \nu, \mu_2 + \nu)$$

Yes, by Kantorovich--Rubinstein duality $W(\mu_1,\mu_2)=\sup_{f\,\text{is 1-Lip}} \int f d(\mu_1-\mu_2)$.