We introduce and study a multi-marginal optimal partial transport problem.Under a natural and sharp condition on the dominating marginals, we establishuniqueness of the optimal plan. Our strategy of proof establishes and exploitsa connection with another novel problem, which we call the Monge-Kantorovichpartial barycenter problem (with quadratic cost). This latter problem has anatural interpretation as a variant of the mines and factories description ofoptimal transport. We then turn our attention to various analytic properties ofthese two problems. Of particular interest, we show that monotonicity of theactive marginals can fail, a surprising difference from the two marginal case.
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