I stumbled upon this interesting statement in this paper: "One interesting observation is that the Wasserstein ambiguity set with the Wasserstein order p = 2 is less conservative, because the 2-Wasserstein distance is stronger than 1-Wasserstein distance, i.e., $W_1(\mu,\nu) \leq W_2(\mu,\nu)$. Hence, the 2-Wasserstein ambiguity set is smaller than the 1-Wasserstein ambiguity set, which leads to less conservative costs."
Here $W_1(\mu,\nu)$ and $W_2(\mu,\nu)$ are the Wasserstein distances of order 1 and 2, respectively, between the distributions $\mu$ and $\nu$.
The Wasserstein distance is given by $$W_p(\mu,\nu):= \inf \left\{\int_{\Xi^2}||\xi_1 - \xi_2||^p \Pi(d\xi_1,d\xi_2) \right\}^{\frac{1}{p}}$$, where $\Pi$ is a joint distirbution of $\xi_1$ and $\xi_2$ with marginals $\mu$ and $\nu$, respectively.
Now to my question: Does the following relation between the $p-$Wasserstein distance hold: $$W_1(\mu,\nu)\leq W_2(\mu,\nu) \leq ... \leq W_\infty(\mu,\nu).$$
My initial thought is that it holds because of Jensen's inequality.
