  Wasserstein projections in the convex order were first considered in the
framework of weak optimal transport, and found application in various problems
such as concentration 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.
