Quantifying the degree of dissimilarity between two probability distributions on a finite metric space X is a fundamental task in Computer Science and Computer Vision. A natural dissimilarity measurement based on optimal transport is the earth mover’s distance (EMD), also known as the Kantorovich metric or Wasserstein-1 metric. We denote the metric space of probability measures on X equipped with the earth mover’s distance as EMD(X), called the earth mover’s space. A key technique for analyzing this metric – pioneered by Charikar and Indyk-Thaper – involves constructing low-distortion embeddings of EMD(X) into the Lebesgue space L1. The best upper bound for the distortion of an embedding of EMD(X) with |X|=n into L1 is O(logn). This result follows from a combination of Charikar’s work (which builds on work of Kleinberg and Tardos) and the seminal result by Fakcharoenphol, Rao, and Talwar. Moreover, it is well known that expander graphs yield a matching lower bound of Ω(logn) for L1-distortion, showing that the upper bound can be tight.