Interior regularity of optimal transport paths
Optimal transport paths arise in one of the many variants of the mass transport problem. Precisely the author consider a mass transport problem for which the cost of transport depends on the actual transport path rather then on the distance between the original and the final location of the mass. In this paper transport paths are real flat 1-chains and the associated cost is given by the \(M^\alpha\)-mass (extended from polyhedral chains) for some \(\alpha \in [0,1]\). In a previous paper the author proved that a real flat \(k\)-chain with finite \(M^\alpha\)-mass is rectifiable. In this paper first a tangent cone property is introduced for optimal transport paths. Then it is proved that near an interior point a local finiteness property holds. It follows that near an interior point and optimal transport path is a finite union of segments.
scientific article; zbMATH DE number 2123231