Generalized Wasserstein Barycenters

We study the existence and uniqueness of the barycenter of a signed distribution of probability measures on a Hilbert space. The barycenter is found, as usual, as a minimum of a functional. In the case where the positive part of the signed measure is atomic, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the L2-barycenter of the quantiles on the cone of nonincreasing functions in L2(0,1). Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in ℝ2. Finally, we address the consistency of the barycenters and we prove that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure.