We do three things. First, we characterize the class of measures in P_2(M) such that for any other measure in 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
On the inverse implication of Brenier-McCann theorems and the structure of P_2(M) / Gigli, Nicola. - In: METHODS AND APPLICATIONS OF ANALYSIS. - ISSN 1073-2772. - 18:2(2011), pp. 127-158.
On the inverse implication of Brenier-McCann theorems and the structure of P_2(M).
Gigli, Nicola
2011-01-01
Abstract
We do three things. First, we characterize the class of measures in P_2(M) such that for any other measure in 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| File | Dimensione | Formato | |
|---|---|---|---|
|
inversebrenier.pdf
non disponibili
Licenza:
Non specificato
Dimensione
531.21 kB
Formato
Adobe PDF
|
531.21 kB | Adobe PDF | Visualizza/Apri Richiedi una copia |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
