Displacement convexity of Entropy and the distance cost Optimal Transportation

Fabio Cavalletti,Flavia Santarcangelo,Nicola Gigli

doi:10.5802/afst.1679

Abstract

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 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 CD 1 (K,N) condition can be formulated in terms of displacement convexity along W 1 -geodesics.

Highlights

The formulation of an appropriate version of Ricci curvature lower bounds valid for possibly singular spaces has been a central topic of research for several years
During the last decade Optimal Transport had a relevant role in the topic that culminated with the successful theory of Lott–Villani [10] and Sturm [15, 16] of metric measure spaces verifying a lower bound on the Ricci curvature in a synthetic sense
A space will satisfy the CD(K, N ) condition if the entropy evaluated along W2-geodesics is more convex than the entropy evaluated along W2-geodesics of the model space with constant curvature K and dimension N in an appropriate sense

Summary

Introduction

The formulation of an appropriate version of Ricci curvature lower bounds valid for possibly singular spaces has been a central topic of research for several years. Motivated by the proof of the local-to-global property for the curvaturedimension condition, in [5] has been shown that a metric measure space (X, d, m) verifies CD(K, N ) if and only if it satisfies CD1(K, N ), provided X is essentially non-branching (see Definition 2.1) and the total space to have finite mass (i.e. m(X) < ∞) It was recently addressed whether or not the CD condition really depends on the special exponent p = 2 used to check displacement convexity of entropy. While for smooth manifold it is clear that it does not (being equivalent to a lower bound on the Ricci tensor) the general case of metric measure spaces has been considered in the recent [1] where complete equivalence will be proved It remained unclear if the CD1(K, N ) condition could be equivalently formulated in terms of displacement convexity of the Entropy functional along W1-geodesics. The CD1(K, N ) condition is formulated, in analogy with the classical CD(K, N ), as displacement convexity of entropy along W1-geodesics; its precise formulation is given in Definition 2.5

Background material

L1-Optimal Transport

Curvature-Dimension conditions

Equivalent Formulations of Ricci Curvature bounds