A proof of Sudakov theorem with strictly convex norms
The paper establishes a solution to the Monge problem in ℝ for a possibly asymmetric norm cost function and absolutely continuous initial measures, under the assumption that the unit ball is strictly convex-but not necessarily differentiable nor uniformly convex. The proof follows the strategy initi...
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| Format: | Journal article |
| Language: | English |
| Published: |
2011
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| Summary: | The paper establishes a solution to the Monge problem in ℝ for a possibly asymmetric norm cost function and absolutely continuous initial measures, under the assumption that the unit ball is strictly convex-but not necessarily differentiable nor uniformly convex. The proof follows the strategy initially proposed by Sudakov in 1976, found to be incomplete in 2000; the missing step is fixed in the above case adapting a disintegration technique introduced for a variational problem. By strict convexity, mass moves along rays, and we also investigate the divergence of the vector field of rays. © 2010 Springer-Verlag. |
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