Monge's transport problem on a Riemannian manifold

Mikhail Feldman, Robert J. McCann
2001 Transactions of the American Mathematical Society  
Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures µ + = µ − , find the measurepreserving map s : M −→ M between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold M -and assuming absolute continuity of µ + -an optimal map will be shown to exist. Aspects of its uniqueness are also established.
doi:10.1090/s0002-9947-01-02930-0 fatcat:gu53d5qzfffgfb5dricxp2kzry