Calculus of Variations and Geometric Measure Theory
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N. Gigli

On the inverse implication of Brenier-McCann theorems and the structure of $(P_2(M),W_2)$

created by gigli on 15 Oct 2009
modified on 02 Mar 2012


Accepted Paper

Inserted: 15 oct 2009
Last Updated: 2 mar 2012

Journal: Meth. Appl. of Anal.
Year: 2009


We do three things. First, we characterize the class of measures $\mu\in mathscr P_2(M)$ such that for any other $\nu\in\mathscr P_2(M)$ there exists a unique optimal transport plan, and this plan is induced by a map. Second, we study the tangent space at any measure and we identify the class of measures for which the tangent space is an Hilbert space. Third, we prove that these two classes of measures coincide. This answers a question recently raised by Villani. Our results concerning the tangent space can be extended to the case of Alexandrov spaces.

Keywords: riemannian, Optimal map, Wasserstein, tangent space


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