6533b856fe1ef96bd12b1c44

RESEARCH PRODUCT

Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces

Tapio RajalaLuigi Ambrosio

subject

Mathematics - Differential GeometryPure mathematicsGeodesicApplied MathematicsInjective metric spacenon-brancingMathematical analysis49Q20 53C23Metric Geometry (math.MG)Measure (mathematics)geodesic metric spaceConvex metric spaceIntrinsic metricMetric spaceMathematics - Metric GeometryDifferential Geometry (math.DG)Metric (mathematics)FOS: Mathematicsupper gradientMetric mapoptimal transportationMathematics

description

We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps.

yearjournalcountryeditionlanguage
2014-01-01
https://dx.doi.org/10.48550/arxiv.1111.5119