  We show that any $L_1$ embedding of the transportation cost (a.k.a.
Earthmover) metric on probability measures supported on the grid
$\{0,1,...,n\}^2\subseteq \R^2$ incurs distortion $\Omega(\sqrt{\log n})$. We
also use Fourier analytic techniques to construct a simple $L_1$ embedding of
this space which has distortion $O(\log n)$.
