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  "path": "/abs/2602.19434v1",
  "publishedAt": "2026-02-24T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Chris Gartland",
    "Mikhail Ostrovskii",
    "Yuval Rabani",
    "Robert Young"
  ],
  "textContent": "**Authors:** Chris Gartland, Mikhail Ostrovskii, Yuval Rabani, Robert Young\n\nWe 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 $Ω(\\log N)$, 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.",
  "title": "$L_1$-distortion of Earth Mover Distances and Transportation Cost Spaces on High Dimensional Grids"
}