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Steiner problems in optimal transport

We study the Steiner problem of finding a minimal spanning network in the setting of a space of probability measures with metric defined by the cost of optimal transport between measures. The existence of a solution is shown for the Wasserstein space P p (χ) over any base space χ which is a separable, locally compact Hadamard space. Structural results are given for the case P 2 (ℝ n ).