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Optimisation under constraint of Wasserstein distance

recently we have a pre-print (https://arxiv.org/abs/2109.11926) that can answer your third question, whether DRO with entropic regularized Wasserstein distance (called the Sinkhorn DRO framework) is ...
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Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map

First a comment: you write (before stating your Theorem 3.14) that "The existence and uniqueness of the solution of the Monge Problem is guaranteed if $c$ is strictly convex and the supports of $\...

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