  We develop a notion of projections between sets of probability measures using
the geometric properties of the 2-Wasserstein space. It is designed for general
multivariate probability measures, is computationally efficient to implement,
and provides a unique solution in regular settings. The idea is to work on
regular tangent cones of the Wasserstein space using generalized geodesics. Its
structure and computational properties make the method applicable in a variety
of settings, from causal inference to the analysis of object data. An
application to estimating causal effects yields a generalization of the notion
of synthetic controls to multivariate data with individual-level heterogeneity,
as well as a way to estimate optimal weights jointly over all time periods.
