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Section: New Results

Optimal transport and sub-Riemannian geometry

Uniquely minimizing costs for the Kantorovitch problem

Participants : Ludovic Rifford, Robert Mccann [Univ of Toronto, Canada] , Abbas Moameni [Carleton Univ, Ottawa, Canada] .

In continuation of the work by McCann and Rifford [65], a paper by Moameni and Rifford [24] study some conditions on the cost which are sufficient for the uniqueness of optimal plans (provided that the measures are absolutely continuous with respect to the Lebesgue measure). As a by-product of their results, the authors show that the costs which are uniquely minimizing for the Kantorovitch problem are dense in the ${C}^{0}$-topology. Many others applications and examples are investigated.

The Sard conjecture in sub-Riemannian geometry, optimal transport and measure contraction properties

Participants : Zeinab Badreddine, Ludovic Rifford.

Zeinab Badreddine [13] obtained the first result of well-posedness for the Monge problem in the sub-Riemannian setting in the presence singular minimizing curves. This study is related to the so-called measure contraction property. In collaboration with Rifford [14], Badreddine obtained new classes of sub-Riemannian structures satisfying measure contraction properties.