We show that any L1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0,1,...,n}2⊆R2 incurs distortion Ω(logn). We also use Fourier analytic techniques to construct a simple L1 embedding of this space which has distortion O(logn).
Cite
@article{arxiv.cs/0509074,
title = {Planar Earthmover is not in $L_1$},
author = {Assaf Naor and Gideon Schechtman},
journal= {arXiv preprint arXiv:cs/0509074},
year = {2007}
}