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

Abstract

We do three things. First, we characterize the class of measures µ ∈ P 2 (M) such that for any other ν ∈ 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. Introduction. Among the several papers devoted to the study of mass transportation problems, two can certainly be called cornerstones of the theory: the work of Brenier [6] (together with the generalization to the case of Riemannian manifolds due to McCann [25]) where existence, uniqueness and structure of the optimal transport map is established, and the work of Otto [28], where the Riemannian structure of the space (P 2 (M), W 2) is described. The theory has been deeply studied in the past years. A topic which became suddenly clear, in particular for what concerns the Riemannian structure of the space of measures, is the fact that there are 'good' measures (like absolutely continuous ones) near which the Riemannian structure behaves nicely, and 'bad' measures (like deltas) at which such structure degenerates. The precise borderline between these two kind of measures was up to now not completely understood, and the question of finding the 'right' structure of the space (P 2 (M), W 2) was also recently posed in Villani's monograph [33]. The problem of the gray area between 'good' measures and 'bad' ones appears also in Brenier-McCann theorems. Indeed, the typical statement of such theorem is: Assume that µ, ν ∈ P 2 (M) are such that µ gives 0 mass to dim(M) − 1 dimensional sets, then there exists a unique optimal transport plan, and such plan is induced by a map (where a structural characterization of the map in terms of Kantorovich potential is also given). Now, the point is that the assumption made on µ, although clearly sufficient to get the conclusion, is not necessary. Given the fundamental importance of the Brenier-McCann theorems, it is natural to look for the sharp hypothesis in their statement.