Planar Earthmover Is Not in $L_1$

Abstract

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