  The purpose of this paper is to provide a systematic discussion of a
generalized barycenter based on a variant of unbalanced optimal transport (UOT)
that defines a distance between general non-negative, finitely supported
measures by allowing for mass creation and destruction modeled by some cost
parameter. They are denoted as Kantorovich-Rubinstein (KR) barycenter and
distance. In particular, we detail the influence of the cost parameter to
structural properties of the KR barycenter and the KR distance. For the latter
we highlight a closed form solution on ultra-metric trees. The support of such
KR barycenters of finitely supported measures turns out to be finite in general
and its structure to be explicitly specified by the support of the input
measures. Additionally, we prove the existence of sparse KR barycenters and
discuss potential computational approaches. The performance of the KR
barycenter is compared to the OT barycenter on a multitude of synthetic
datasets. We also consider barycenters based on the recently introduced
Gaussian Hellinger-Kantorovich and Wasserstein-Fisher-Rao distances.
