Displacement convexity of Entropy and the distance cost Optimal Transportation

Cavalletti, Fabio; Gigli, Nicola; Santarcangelo, Flavia

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 2, pp. 411-427.

Pendant la dernière décennie le Transport Optimal a eu un rôle remarquable dans l’étude de la géométrie des espaces singulièrs qui a culminé dans la théorie de Lott–Sturm–Villani. Cette dernière repose sur la caractèrisation des bornes inférieures de la courbure de Ricci en termes de convexité de déplacement de la fonctionnelle entropie le long des $W_2$-géodésiques. Récentes avancées dans la théorie (technique de localisation et local-au-global propriété) ont été obtenus en envisageant le différent point de vue du $L^1$ Transport Optimal, en entraînant à la differénte condition de courbure-dimension ${\mathsf{CD}}^1(K,N)$ [5]. Cette dernière est formulée en termes des propriétés $1$-dimensionnelles de la courbure des courbes integralés associées aux fonctions lipschitziennes. Dans la présente note on prouve que les deux approches produisent la même condition de courbure-dimension, en conciliant les deux définitions. En particulier, on prouve que la condition ${\mathsf{CD}}^1(K,N)$ peut être formulée en termes de convexité de déplacement le long des $W_1$-géodésiques.

During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott–Sturm–Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in terms of displacement convexity of certain entropy functionals along $W_2$-geodesics. Substantial recent advancements in the theory (localization paradigm and local-to-global property) have been obtained considering the different point of view of $L^1$-Optimal transport problems yielding a different curvature dimension ${\mathsf{CD}}^1(K,N)$ [5] formulated in terms of one-dimensional curvature properties of integral curves of Lipschitz maps. In this note we show that the two approaches produce the same curvature-dimension condition reconciling the two definitions. In particular we show that the ${\mathsf{CD}}^1(K,N)$ condition can be formulated in terms of displacement convexity along $W_1$-geodesics.

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