Regularity for a log-concave to log-concave mass transfer problem with near Euclidean cost

If the cost function is not too far from the Euclidean cost, then the optimal map transporting Gaussians restricted to a ball will be regular. Similarly, given any cost function which is smooth in a neighborhood of two points on a manifold, there are small neighborhoods near each such that a Gaussian restricted to one is transported smoothly to a Gaussian on the other.

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Published 20 June 2011