On Interpolation and Curvature via Wasserstein Geodesics

Abstract

In this article, a proof of the interpolation inequality along geodesics in $p$-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli's recently developed calculus on metric measure spaces, even a $q$-Laplacian comparison theorem holds on $q$-infinitesimal convex spaces. In the appendix, the theory of Orlicz-Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

35+14 pages, comments are welcome, added remark on relation between weak and strong curvature condition

References in corpus (8)

• The splitting theorem in non-smooth context
• Li-Yau and Harnack type inequalities in $RC{D}^{\ast }(K,N)$ metric measure spaces
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Harmonic functions on metric measure spaces
• $(K,N)$-convexity and the curvature-dimension condition for negative $N$
• Hamilton Jacobi equations on metric spaces and transport-entropy inequalities
• Stability of the global attractor under Markov-Wasserstein noise
• Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below

Cited by in corpus (8)

• An optimal transport formulation of the Einstein equations of general relativity
• Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
• On quotients of spaces with Ricci curvature bounded below
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• On proximal mappings with Young functions in uniformly convex Banach spaces
• Kurdyka-Lojasiewicz-Simon inequality for gradient flows in metric spaces
• On the curvature and heat flow on Hamiltonian systems
• On nonexpansiveness of metric projection operators on Wasserstein spaces