Recently, Liero, Mielke and Savaré introduced the Hellinger–Kantorovich distance on the space of nonnegative Radon measures of a metric space X. We prove that Hellinger–Kantorovich barycenters always exist for a class of metric spaces containing of compact spaces and Polish \(\mathrm{CAT}(1)\) spaces; and if we assume further some conditions on the data, such barycenters are unique. We also introduce homogeneous multimarginal problems and illustrate some relations between their solutions and Hellinger–Kantorovich barycenters. Our results are analogous to the work of Agueh and Carlier for Wasserstein barycenters.

最近，Liero，Mielke和Savaré在度量空间X的非负Radon度量空间上引入了Hellinger-Kantorovich距离。我们证明了对于包含紧凑空间和波兰\（\ mathrm {CAT}（1）\）空间的一类度量空间，Hellinger–Kantorovich重心始终存在；如果我们进一步假设数据的某些条件，则此类重心是唯一的。我们还将介绍齐次边际问题，并说明它们的解与Hellinger-Kantorovich重心之间的一些关系。我们的结果类似于Agueh和Carlier为Wasserstein重心所做的工作。