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Uniqueness and transport density in Monge’s mass transportation problem. (English) Zbl 1003.49031

Summary: Monge’s problem refers to the classical problem of optimally transporting mass: given Borel probability measures \(\mu^+ \neq \mu^-\) on \({\mathbb{R}}^n\), find the measure preserving map \(s(x)\) between them which minimizes the average distance transported. Here distance can be induced by the Euclidean norm, or any other uniformly convex and smooth norm \(d(x,y) = \| x - y\|\) on \({\mathbb{R}}^n\). Although the solution is never unique, we give a geometrical monotonicity condition singling out a particular optimal map \(s(x)\). Furthermore, a local definition is given for the transport cost density associated to each optimal map. All optimal maps are then shown to lead to the same transport density \(a \in L^1({\mathbb{R}}^n)\).

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
28A50 Integration and disintegration of measures
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