Planar Earthmover is not in L_1

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
sube Ropf
²
incurs distortion Omega(radic;(log n)). We also use Fourier analytic techniques to construct a simple L
₁
embedding of this space which has distortion O(log n)