  The most challenging scenario for Kohn-Sham density functional theory, that
is when the electrons move relatively slowly trying to avoid each other as much
as possible because of their repulsion (strong-interaction limit), is
reformulated here as an optimal transport (or mass transportation theory)
problem, a well established field of mathematics and economics. In practice, we
show that solving the problem of finding the minimum possible internal
repulsion energy for $N$ electrons in a given density $\rho(\rv)$ is equivalent
to find the optimal way of transporting $N-1$ times the density $\rho$ into
itself, with cost function given by the Coulomb repulsion. We use this link to
put the strong-interaction limit of density functional theory on firm grounds
and to discuss the potential practical aspects of this reformulation.
