  We study the barycenter of the Hellinger--Kantorovich metric over
non-negative measures on compact, convex subsets of $\mathbb{R}^d$. The article
establishes existence, uniqueness (under suitable assumptions) and equivalence
between a coupled-two-marginal and a multi-marginal formulation. We analyze the
HK barycenter between Dirac measures in detail, and find that it differs
substantially from the Wasserstein barycenter by exhibiting a local
`clustering' behaviour, depending on the length scale of the input measures. In
applications it makes sense to simultaneously consider all choices of this
scale, leading to a 1-parameter family of barycenters. We demonstrate the
usefulness of this family by analyzing point clouds sampled from a mixture of
Gaussians and inferring the number and location of the underlying Gaussians.
