Planar Earthmover is not in $L_1$

Assaf Naor; Gideon Schechtman

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arXiv · cs/0509074

Planar Earthmover is not in $L_1$

Assaf Naor·Gideon Schechtman

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Abstract

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

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Assaf Naor, Gideon Schechtman. 2005-09-26. Planar Earthmover is not in $L_1$. https://arxiv.org/abs/cs/0509074

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