Search results for "optimal transportation"

showing 2 items of 2 documents

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

2014

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.

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
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On deterministic solutions for multi-marginal optimal transport with Coulomb cost

2022

In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane $\R^2$. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.

Multimarginal optimal transportation Monge-Kantorovich problem Duality theory Coulomb cost Density Functional Theory.Applied MathematicstiheysfunktionaaliteoriaFOS: Physical sciencesMonge-Kantorovich problemduality theoryvariaatiolaskentaMathematical Physics (math-ph)General MedicineDensity Functional Theory.matemaattinen optimointimultimarginal optimal transportation49J45 49N15 49K30Mathematics - Analysis of PDEsOptimization and Control (math.OC)Coulomb costFOS: MathematicsMathematics - Optimization and ControlMathematical PhysicsAnalysisAnalysis of PDEs (math.AP)
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