  In this paper, we introduce a generalization of the Wasserstein barycenter,
to a case where the initial probability measures live on different subspaces of
R^d. We study the existence and uniqueness of this barycenter, we show how it
is related to a larger multi-marginal optimal transport problem, and we propose
a dual formulation. Finally, we explain how to compute numerically this
generalized barycenter on discrete distributions, and we propose an explicit
solution for Gaussian distributions.
