I am reading the paper Barycenters in the Wasserstein space by Martial Agueh and Guillaume Carlier. In Proposition 3.5, they prove the existence and uniqueness of $$\nu \mapsto \sum_{i=1}^p \frac{\lambda_i}{2}W_2^2(\nu_i, \nu).$$
There is no mention of uniqueness in the proof. Why is the minimizer of $\nu \mapsto \sum_{i=1}^p \frac{\lambda_i}{2}W_2^2(\nu_i, \nu)$ unique?
Uniqueness follows directly from proposition 3.3 on the previous page. For completeness, a shortened version containing the relevant parts:
Suppose $\mu$ and $\nu$ are probability measures on $\mathbb{R}^d$ with finite second moment and $\mu$ does not give mass to small sets. There exists a $\mu$-almost surely unique $\nabla \phi: \mathbb{R}^d \to \mathbb{R}^d$, which is the gradient of a convex function, such that $$\pi = \mu \circ (\mathrm{Id}\times \nabla\phi)^{-1}$$ is the unique optimal transportation plan for the quadratic cost. The resulting coupling $\mu \circ \nabla \phi^{-1} = \nu$ must then be optimal.
This is also known as Brenier's theorem (or more precisely McCann's generalization thereof); for a textbook reference see theorem 2.12 (ii) of [1].
Leading up to proposition 3.5, we've already done most of the hard work to prove uniqueness. In some sense, section 2 and 3 of the paper constitute one long proof, via duality, that for each $\nu_i$ there is a suitable convex map $\phi_i$. Assuming that one of the $\nu_i$ does not give mass to small sets is the final piece of the puzzle and everything falls into place via Brenier's theorem.
[1] Villani, Cédric, Topics in optimal transportation, Graduate Studies in Mathematics 58. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3312-X/hbk). xvi, 370 p. (2003). ZBL1106.90001.
I agree with the answer by @Gabriel Clara in principle. However I feel it is not complete clear in their proof. They basically wanted to rely on the uniqueness from Brenier map between one marginal $\mu_i$ (that is absolutely continuous) and the barycenter, to show the uniqueness of the barycenter.
In detail, they wanted to show that for any given solution $\nu$ to the primal problem (called as barycenter), it can be always defined by the pushforward of $\mu_i$. As the Brenier map is unique, thus $\nu$ is unique. However, they assume those dual solutions do not depend on $\nu$, which is true at the stage when they assume. But to apply the conclusion from (3.3) and (3.4):
\begin{align} &\textrm{(3.3)}\qquad \frac{\lambda_i}{2}|x-y|^2 = S_{\lambda_i}f_i(x)+f_i(y),\qquad \gamma_i-a.e.\\ &\textrm{(3.4)}\qquad \sum S_{\lambda_i}(S_{\lambda_i}f_i)\geq 0,\qquad \sum S_{\lambda_i}(S_{\lambda_i}f_i)=0,\quad \nu-a.e. \end{align} they use "for any dual solution, there exists some $\nu$ is a primal solution s.t" \begin{align} \frac{\lambda_i}{2}W_2^2(\nu_i,\nu)=\int S_{\lambda_i}f_i\mathrm{d}\nu_i + \int f_i\mathrm{d}\nu, \end{align} where it seems hard to deny the dependence between the primal solution $\nu$ and the dual solutions.
