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Geometric interpolation inequalities: from Riemannian to sub-Riemannian geometry

Davide Barilari
Thursday, November 23, 2017 - 11:30

In a seminal paper of Cordero-Erasquin-McCann-Schmuckenschläger, the authors extends using optimal transport techniques classical functional and geometrical interpolation inequalities from the Euclidean to the Riemannian setting. In particular these results imply a "geodesic" version of the celebrated Brunn-Minkovski inequality.

Sub-Riemannian manifolds can be described as limits of Riemannian ones with Ricci going to $-\infty$ and the generalization of the above results is not possible using classical theory of Riemannian curvature bounds.

In this talk, we discuss how, under generic assumptions, these structures support interpolation inequalities. As a byproduct, we characterize the sub-Riemannian cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex. The techniques are based on sub-Riemannian optimal transport and Jacobi fields.
[Joint work with Luca Rizzi] 

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