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arXiv · 2602.19434

$L_1$-distortion of Earth Mover Distances and Transportation Cost Spaces on High Dimensional Grids

Chris Gartland·Mikhail Ostrovskii·Yuval Rabani·Robert Young
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Abstract

We prove that the distortion of any embedding into $L_1$ of the transportation cost space or earth mover distance over a $d$-dimensional grid $\{1,\dots m\}^d$ is $\Omega(\log N)$, where $N$ is the number of vertices and the implicit constant is universal (in particular, independent of dimension). This lower bound matches the universal upper bound $O(\log N)$ holding for any $N$-point metric space. Our proof relies on a new Sobolev inequality for real-valued functions on the grid, based on random measures supported on dyadic cubes.

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Chris Gartland, Mikhail Ostrovskii, Yuval Rabani, Robert Young. 2026-02-23. $L_1$-distortion of Earth Mover Distances and Transportation Cost Spaces on High Dimensional Grids. https://arxiv.org/abs/2602.19434

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