# 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

## 1 Answer

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 .

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.

 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.