  Recently, Liero, Mielke and Savar\'{e} introduced Hellinger-Kantorovich
distance on the space of nonnegative Radon measures of a metric space $X$
[19,20]. We prove that Hellinger-Kantorovich barycenters always exist for a
class of metric spaces containing of compact spaces, and Polish $CAT(1)$
spaces; and if we assume further some conditions on starting measures, such
barycenters are unique. We also introduce homogeneous multimarginal problems
and illustrate some relations between their solutions with
Hellinger-Kantorovich barycenters. Our results are analogous to the work of
Agueh and Carlier [1] for Wassertein barycenters.
