  Wasserstein barycenters define averages of probability measures in a
geometrically meaningful way. Their use is increasingly popular in applied
fields, such as image, geometry or language processing. In these fields
however, the probability measures of interest are often not accessible in their
entirety and the practitioner may have to deal with statistical or
computational approximations instead. In this article, we quantify the effect
of such approximations on the corresponding barycenters. We show that
Wasserstein barycenters depend in a H{\"o}lder-continuous way on their
marginals under relatively mild assumptions. Our proof relies on recent
estimates that quantify the strong convexity of the dual quadratic optimal
transport problem and a new result that allows to control the modulus of
continuity of the push-forward operation under a (not necessarily smooth)
optimal transport map.
