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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 (√log n). We also use Fourier analytic techniques to construct a simple L₁ embedding of this space which has distortion O(log n).
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Naor et al. (2007) studied this question.
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