Lipschitz continuity of the Wasserstein projections in the convex order on the line

Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found applications in various problems such as con-centration inequalities and martingale optimal transport. In dimension one, it is well-known that the set of probability measures with a given mean is a lattice w.r.t. the convex order. Our main result is that, contrary to the minimum and maximum in the convex order, the Wasserstein projections are Lipschitz continuity w.r.t. the Wasserstein distance in dimension one. Moreover, we provide examples that show sharpness of the obtained bounds for the 1-Wasserstein distance.