I am reading the paper [Barycenters in the Wasserstein space](https://doi.org/10.1137/100805741) 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?



The uniqueness of Barycenters in the Wasserstein space
Uniqueness follows directly from proposition 3.3 on the previous page. For completeness, a shortened version containing the relevant parts:



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> 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.
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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](https://zbmath.org/?q=an:1106.90001).


