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?

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


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