On the inverse implication of Brenier-Mccann theorems and the structure of (P 2 (M),W 2 )

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BV Estimates in Optimal Transportation and Applications (2015, background): “(d − 1)−dimensional); see [14] for the sharp assumptions and notions about this issue.”
A User’s Guide to Optimal Transport (2013, background): “1 has been proved in [46].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “Gigli in [183] to prove the following characterization theorem, extending 1.”
Optimal Maps and Exponentiation on Finite-Dimensional Spaces with Ricci Curvature Bounded from Below (2013, unknown): “We shall also use the following lemma, whose proof was given in [11] (see also [12]) for the case of Riemannian manifolds; yet, the argument is only metric and can be repeated without any change.”
Statistical Aspects of Wasserstein Distances (2018, background): “See Gigli (2011) for a description of the tangent bundle when the underlying space R d is replaced by a Riemannian manifold.”