  In this paper, for $\mu$ and $\nu$ two probability measures on $\mathbb{R}^d$
with finite moments of order $\rho\ge 1$, we define the respective projections
for the $W_\rho$-Wasserstein distance of $\mu$ and $\nu$ on the sets of
probability measures dominated by $\nu$ and of probability measures larger than
$\mu$ in the convex order. The $W_2$-projection of $\mu$ can be easily computed
when $\mu$ and $\nu$ have finite support by solving a quadratic optimization
problem with linear constraints. In dimension $d=1$, Gozlan et al.~(2018) have
shown that the projections do not depend on $\rho$. We explicit their quantile
functions in terms of those of $\mu$ and $\nu$. The motivation is the design of
sampling techniques preserving the convex order in order to approximate
Martingale Optimal Transport problems by using linear programming solvers. We
prove convergence of the Wasserstein projection based sampling methods as the
sample sizes tend to infinity and illustrate them by numerical experiments.
