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 $Ω(\log N)$, 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.

翻译：我们证明，对于$d$维网格$\{1,\dots m\}^d$上的运输成本空间或地球移动距离，其嵌入$L_1$空间的失真度下界为$Ω(\log N)$，其中$N$为顶点数量，且隐含常数是普适的（尤其与维度无关）。该下界与适用于任意$N$点度量空间的普适上界$O(\log N)$相匹配。我们的证明依赖于基于二进立方体上随机测度的网格实值函数新Sobolev不等式。