In this talk, I will present a new ansatz space for the general symmetric multimarginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from combinatorial in both N and `l to `l · (N + 1), where `l 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 `l · (N - 1) unknowns, and cures the insuffciency of the Monge ansatz, i.e. it is shown 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.

These results were established in collaboration with Gero Friesecke. The corresponding paper can be found under arXiv:1801.00341.