Hellinger–Kantorovich barycenter between Dirac measures

The Hellinger-Kantorovich (HK) distance is an unbalanced extension of the Wasserstein-2 distance. It was shown recently that HK barycenter exhibits a much more complex behaviour than Wasserstein barycenter. Motivated by this observation we study in detail for case where input measures are uncountable collection Dirac measures, particular dependency on length scale parameter HK, question whether discrete or continuous and relation between expected empirical analytical results complemented with numerical experiments demonstrate can provide coarse-to-fine representation pointcloud measure.