In Optimal transportation, and more precisely in "Optimal Transport: Old and New" (Definition. 6.1, on page 106 - actually on page 111 out of 998, in this link), the Wasserstein distance $W_p$ is defined through the so-called "cost function", $c(x,y)=d(x,y)^{p}$, where $p\in [1,\infty)$:
$W_{p}(\mu,\nu) = \Bigg( \inf_{\pi \in \Pi(\mu,\nu)} \int_{X} d(x,y)^{p} d\pi(x,y) \Bigg) ^{\frac{1}{p}}$
I cannot find a clear definition of the object $d(x,y)^{p}$, that should be, likely, an Euclidean distance (maybe, something like $d(x,y)^{p}=\left|x-y\right|^{p}$ ?). Therefore, what is the conventional definition of $d(x,y)^{p}$?