Calculus of Variations and Geometric Measure Theory
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F. Cavalletti - N. Gigli - F. Santarcangelo

Displacement convexity of Entropy and the distance cost Optimal Transportation

created by cavallett on 01 May 2020
modified on 24 Feb 2021


Accepted Paper

Inserted: 1 may 2020
Last Updated: 24 feb 2021

Journal: Annales de la faculté des sciences de Toulouse
Year: 2020


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 $\mathsf{CD}^{1}(K,N)$ 8 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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