Methods and Applications of Analysis

Volume 18 (2011)

Number 2

On the inverse implication of Brenier-Mccann theorems and the structure of (P2(M),W2)

Pages: 127 – 158



Nicola Gigli (Department of Mathematics, University of Nice, France)


We do three things. First, we characterize the class of measures $μ ∈ P2(M)$ such that for any other $ν ∈ P2(M)$ 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.


optimal transport map, tangent space

2010 Mathematics Subject Classification


Published 23 September 2011