What is the title inequality and why is it true?
For context: in the paper I'm reading, this inequality is cited to show that for any measures $\mu, \nu$, $$\int |x|^2 d\mu \leq 2 \int |x|^2 d\nu + 2 W_2(\mu,\nu)^2$$
where $W_2(\mu,\nu)^2 = \inf_{p(\mu,\nu)} \int |x-y|^2 p(dxdy)$ is the Wasserstein distance, and where $p(\mu,\nu)$ is any probability measure with marginals $\mu, \nu$.
I guess $||\mu||^2 = \int |x|^2 d\mu$ represents the $W_2$ distance between $\mu$ and the dirac measure at $0$, and the space of ($L^2$-integrable) probability measures becomes a normed inner product space. Is this a natural way to define the norm of a measure?