Pairwise multi-marginal optimal transport couplings achieve finite constant-factor ratios only for n=1 or snowflake costs with q<1, with sharp Θ(n^{q/2}) dimension growth on R^n and O(√n log s) distortion on grids.
On a class of multidimensional optimal transportation problems,
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Pairwise Multi-marginal Optimal Transport and Embedding for Earth Mover's Distance
Pairwise multi-marginal optimal transport couplings achieve finite constant-factor ratios only for n=1 or snowflake costs with q<1, with sharp Θ(n^{q/2}) dimension growth on R^n and O(√n log s) distortion on grids.