## Abstract

We present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on fnite state spaces which reduces the number of unknowns from (^{N+ℓ1-1})to (N +1), where is the number of marginal states and N the number of marginals. The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to (N 1) unknowns, and cures the insufciency of the Monge ansatz; i.e., we show that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions. Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which has received much recent interest due to its emergence as the strongly correlated limit of Hohenberg-Kohn density functional theory. In this context N corresponds to the number of particles, motivating the interest in large N.

Original language | English |
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Pages (from-to) | 3996-4019 |

Number of pages | 24 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - 2018 |

## Keywords

- Density functional theory
- Optimal transport
- Sparse