From optimal transportation to optimal teleportation

In case the support of μ is disconnected (corresponding to d=∞) we obtain sharp estimates for q=1/p (“optimal teleportation”):limϵ→0⁡ϵ−1/pWp(μ,μ+ϵν)=‖ν‖μ where ‖ν‖μ is expressed in terms of optimal transport on a metric graph, determined only by the relative distances between the connected components of the support of μ, and the weights of the measure ν in each connected component of this support.