Uniqueness and transport density in Monge's mass transportation problem |
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Authors: | Mikhail Feldman Robert J McCann |
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Institution: | (1) Department of Mathematics, University of Wisconsin, Madison, Wisconsin USA 53706 (e-mail: feldman@math.wisc.edu) , US |
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Abstract: | Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures on , 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 on . 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 .
Received: 18 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001 |
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Keywords: | Mathematics Subject Classification (2000): 49Q20 28A50 |
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