# The uniqueness of Barycenters in the Wasserstein space

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?

• The question should be self contained. What are $\nu$, $p$, $\lambda_i$, $W_2$, and $\nu_i$? – LSpice Nov 3 '20 at 2:30
• When you accept an answer, it is not automatically upvoted, so it would probably be polite to upvote it, too. – LSpice Nov 3 '20 at 19:50

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.