Is the Wasserstein metric scale invariant?

Question

Let $X,Y$ be $n$-dimensional random vectors.

Then, is $W(cX,cY)=W(X,Y)$?

Answer 1

Recall that for $1\le p<\infty$, $$ W_p(X,Y)=\left(\inf_{\gamma\in \Gamma(\mu_X,\mu_Y)}\mathsf{E}_{\gamma}[d(X,Y)]^p\right)^{1/p}, $$ where $\mu_X$ and $\mu_Y$ are the distributions of $X$ and $Y$ and $\Gamma$ is the set of all couplings of $\mu_X$ and $\mu_Y$. It is clear that the scale invariance of $W$ depends on the properties of $d(\cdot,\cdot)$. For example, if $d$ is a norm induced metric, $d(cx,cy)=|c|d(x,y)$, and, therefore, $$ W_p(cX,cY)=|c|W_p(X,Y). $$