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On the inverse implication of Brenier-McCann theorems and the structure of $$(\mathcal P_{2}(M),W_{2})$$. (English) Zbl 1284.49050
In this paper, the author studies the mass transportation problem. Recall that in [Y. Brenier, Commun. Pure Appl. Math. 44, No. 4, 375–417 (1991; Zbl 0738.46011)], the existence, uniqueness and structure of the transport map is established. In [R. J. McCann, Geom. Funct. Anal. 11, No. 3, 589–608 (2001; Zbl 1011.58009)], the results have been extended to the case of Riemannian manifolds. Remark also the work [F. Otto, Commun. Partial Differ. Equations 26, No. 1–2, 101–174 (2001; Zbl 0984.35089)], where the Riemannian structure of the space $$(\mathcal{P}_2(M),W_2)$$ is described.
The results are given in the following Brenier-McCann Theorem (Theorem 1.10). Let $$\mu, \nu\in \mathcal{P}_c(M)$$ and assume that $$\mu$$ is absolutely continuous. Then, there exists a unique optimal plan from $$\mu$$ to $$\nu$$ and this plan is induced by the map $$\exp(-\nabla \varphi)$$, where $$\varphi$$ is a Kantorovich potential for $$\mu, \nu$$. The author obtains some results concerning the aspects described above. First, a characterization of those measures to which the Brenier-McCann theorem applies is obtained. Proposition 2.4. Let $$\mu \in \mathcal{P}_2(\mathbb{R}^d)$$. Then for every $$\nu \in \mathcal{P}_2(\mathbb{R}^d)$$ there exists only one optimal plan from $$\mu$$ to $$\nu$$ and this plan is induced by a map from $$\mu$$ if and only if $$\mu$$ is regular. This result is extended to the case of Riemannian manifolds. Then the author obtains a result (Theorem 5.5) giving the identification of the tangent spaces at any measure $$\mu$$. Finally the author proves that the class of measures for which the tangent space is a Hilbert space coincides with the class of measures to which the Brenier-McCann theorem applies. In this case, the tangent space is naturally identified with the well known space of gradients.

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting
##### Keywords:
optimal transport map; Riemannian manifolds; tangent space
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