  We show that the problem of finding the barycenter in the
Hellinger-Kantorovich setting admits a least-cost soft multi-marginal
formulation, provided that a one-sided hard marginal constraint is introduced.
The constrained approach is then shown to admit a conic multi-marginal
reformulation based on defining a single joint multi-marginal perspective cost
function in the conic multi-marginal formulation, as opposed to separate
two-marginal perspective cost functions for each two-marginal problem in the
coupled-two-marginal formulation, as was studied previously in literature. We
further establish that, as in the Wasserstein metric, the recently introduced
framework of unbalanced multi-marginal optimal transport can be reformulated
using the notion of the least cost. Subsequently, we discuss an example when
input measures are Dirac masses and numerically solve an example for Gaussian
measures. Finally, we also explore why the constrained approach can be seen as
a natural extension of a Wasserstein space barycenter to the unbalanced
setting.
