  In this paper, we prove a structure theorem for discrete optimal
transportation plans. We show that, given any pair of discrete probability
measures and a cost function, there exists an optimal transportation plan that
can be expressed as the sum of two deterministic plans. As an application, we
estimate the infinity-Wasserstein distance between two discrete probability
measures $\mu$ and $\nu$ with the $p$-Wasserstein distance, times a constant
depending on $\mu$, $\nu$, and the fixed cost function.
