  We present a new ansatz space for the general symmetric multi-marginal
Kantorovich optimal transport problem on finite state spaces which reduces the
number of unknowns from $\tbinom{N+\ell-1}{\ell-1}$ to $\ell\cdot(N+1)$, where
$\ell$ 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 $\ell\cdot(N-1)$ unknowns, and cures the
insufficiency 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 $N$.

Analysis of PDEs;Quantum Physics