  This work studies the quantitative stability of the quadratic optimal
transport map between a fixed probability density $\rho$ and a probability
measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density
$\rho$ is bounded from above and below on a compact convex set, we prove that
the map $\mu$ $\rightarrow$ T$\mu$ is bi-H{\"o}lder continuous on large
families of probability measures, such as the set of probability measures whose
moment of order p > d is bounded by some constant. These stability estimates
show that the linearized optimal transport metric W2,$\rho$($\mu$, $\nu$) =
T$\mu$ -- T$\nu$ L 2 ($\rho$,R d) is bi-H{\"o}lder equivalent to the
2-Wasserstein distance on such sets, justifiying its use in applications.
