Multi-marginal optimal transport: Theory and applications∗
Department of Mathematical and Statistical Sciences, 632 CAB, University of
Alberta, Edmonton, Alberta, T6G 2G1, Canada.
Over the past five years, multi-marginal optimal transport, a generalization of the well known optimal transport problem of Monge and Kantorovich, has begun to attract considerable attention, due in part to a wide variety of emerging applications. Here, we survey this problem, addressing fundamental theoretical questions including the uniqueness and structure of solutions. The answers to these questions uncover a surprising divergence from the classical two marginal setting, and reflect a delicate dependence on the cost function, which we then illustrate with a series of examples. We go on to describe some applications of the multi-marginal optimal transport problem, focusing primarily on matching in economics and density functional theory in physics.
Mathematics Subject Classification: 49K30 / 49J30 / 49K20 / 91B68 / 81V45 / 90C05 / 35J96
Key words: Multi-marginal optimal transport / Monge−Kantorovich problem / structure of solutions / uniqueness of solutions / matching / purity / density functional theory / strictly correlated electrons
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