A Proofs Proof of Proposition 1. For all x, y R

Thus, (14) is an equality, and u attains the maximum in (5), i.e., it is an optimal dual potential.Proof of Proposition 2. Proof of Proposition 3. We split the proof into 4 parts. Assume the contrary, i.e., that there exist m = m Thus, this case is not possible. Thus, the second case is also not possible. Proof of Proposition 4. We compute W In this section, we provide the details of the training of the OT solvers that we consider. In the images case, the batch size is 32.