Monge’s transport problem on a Riemannian manifold

Feldman, Mikhail; McCann, Robert

doi:10.1090/S0002-9947-01-02930-0

Monge’s transport problem on a Riemannian manifold
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by Mikhail Feldman and Robert J. McCann PDF

Trans. Amer. Math. Soc. 354 (2002), 1667-1697

Abstract:

Monge’s problem refers to the classical problem of optimally transporting mass: given Borel probability measures $\mu ^+ \ne \mu ^-$, find the measure-preserving map $s:M \longrightarrow M$ between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold $M$ — and assuming absolute continuity of $\mu ^+$ — an optimal map will be shown to exist. Aspects of its uniqueness are also established.

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