On displacement interpolation of measures involved in Brenier’s theorem

Juillet, Nicolas

doi:10.1090/S0002-9939-2011-10891-8

On displacement interpolation of measures involved in Brenier’s theorem
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Proc. Amer. Math. Soc. 139 (2011), 3623-3632 Request permission

Abstract:

We prove that in the Wasserstein space built over $\mathbb {R}^d$ the subset of measures that does not charge the non-differentiability set of convex functions is not displacement convex. This completes the study of Gigli on the geometric structure of measures meeting the sharp hypothesis of the refined version of Brenier’s Theorem.

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