  We study the problem of minimizing the Wasserstein distance between a
probability distribution and an algebraic variety. We consider the setting of
finite state spaces and describe the solution depending on the choice of the
ground metric and the given distribution. The Wasserstein distance between the
distribution and the variety is the minimum of a linear functional over a union
of transportation polytopes. We obtain a description in terms of the solutions
of a finite number of systems of polynomial equations. The case analysis is
based on the ground metric. A detailed analysis is given for the two bit
independence model.
