I am trying to understand this 2011 paper by Agueh & Carlier [https://www.ceremade.dauphine.fr/~carlier/AC_bary_Aug11_10.pdf] where they introduce the notion of barycenter in the 2-Wasserstein space, and I am struggling with one of the first points they make in their proof that the problem
$$J(\nu) = \sum\limits_{i = 1}^{p} \frac{\lambda}{2} W_2^2(\nu, \nu_i)$$
where $(\lambda_1, \dots, \lambda_p)$ is a simplex point and $\{\nu_1, \dots, \nu_p\}$ a fixed family of absolutely continuous probability distributions with finite second moments admits a minimizer. Most of the proof I can follow... except the first line, which asserts that it is "easy to check" that if $\{\nu_n\}$ is a minimizing sequence for the functional, then the sequence $$\{\int |x|^2 d\nu_n\}$$ is itself bounded. They suggest using the Kantorovich duality to show this, $$\frac{\lambda}{2}W_2^2(\nu_n, \nu) = \sup \{\int f d\nu_n + \int g d\nu \} = \sup \{\int f d\nu_n + \int \inf_x\{\frac{\lambda}{2}|x - y|^2 - f(x)\} d\nu\}$$
with the $\sup$ being taken over $ f, g \in C_b, f + g \le \frac{\lambda}{2}|x - y|^2$, but I am having trouble proceeding this way. Does anyone know how to see this? The only thing I have been able to work out is that the second moments bound the Wasserstein distance, not the other way around, which is what we need.
