# Upper bounding Wasserstein metric with expectation distance?

Let $$p \geq1$$ and consider space $$\mathcal{P}(\mathbb{R})$$ of Borel probability measures on $$\mathbb{R}$$. The $$p$$-order Wasserstein distance is defined by $$W_p(\mu, \nu) = \left( \inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb{R}^2} |x-y|^p d\pi(x,y) \right)^{1/p},$$ where $$\Pi(\mu,\nu)$$ denotes the set of all Borel probability measures on $$\mathbb{R}^2$$ with marginals $$\mu$$ and $$\nu$$ in $$\mathcal{P}(\mathbb{R})$$.

Let's define the expectation distance between $$\mu$$ and $$\nu$$ by

$$d(\mu,\nu) = \bigg|\int_{\mathbb{R}} x d\mu(x) - \int_{\mathbb{R}} x d\nu(x)\bigg|.$$

While it is easy to prove that $$d(\mu, \nu) \leq W_p(\mu, \nu), \forall \mu,\nu \in \mathcal{P}(\mathbb{R})$$ (e.g., see https://math.stackexchange.com/a/2269491/253451), is there any existing result about upper bounding $$W_p$$ by $$d$$? E.g., Does there exist $$C > 1$$ such that $$W_p(\mu, \nu) \leq C d(\mu,\nu), \forall \mu,\nu \in \mathcal{P}(\mathbb{R})$$? Any counterexample?

$$W_2\left(\mathcal{N}(m_1, \sigma_1), \mathcal{N}(m_2, \sigma_2) \right) = \sqrt{(m_1 - m_2)^2 + (\sigma_1 - \sigma_2)^2}. \\ d\left(\mathcal{N}(m_1, \sigma_1), \mathcal{N}(m_2, \sigma_2) \right) = |m_1 - m_2|.$$
Thus, $$W_2\left(\mathcal{N}(m_1, \sigma_1), \mathcal{N}(m_2, \sigma_2) \right)$$ is arbitrarily larger than $$d\left(\mathcal{N}(m_1, \sigma_1), \mathcal{N}(m_2, \sigma_2) \right)$$; thus such $$C$$ does not exist.