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On the optimal map in the $ 2 $-dimensional random matching problem

L. Ambrosio acknowledges the support of the MIUR PRIN 2015 project. F. Glaudo has received funding from the European Research Council under the Grant Agreement No 721675. D. Trevisan is a member of INdAM GNAMPA group

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  • We show that, on a $ 2 $-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $ L^2 $-norm by identity plus the gradient of the solution to the Poisson problem $ - {\Delta} f^{n, t} = \mu^{n, t}-1 $, where $ \mu^{n, t} $ is an appropriate regularization of the empirical measure associated to the random points. This shows that the ansatz of [8] is strong enough to capture the behavior of the optimal map in addition to the value of the optimal matching cost.

    As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold.

    Mathematics Subject Classification: Primary: 60D05; Secondary: 49J55, 58J35, 35F21.


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  • Figure 1.  The points considered in the proof of of Lemma 4.1

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