The sliced Wasserstein deficit is zero if and only if the Brenier map is homothetic affine, with quantitative stability via the sliced Poincaré-Korn constant of the source measure.
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Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit
The sliced Wasserstein deficit is zero if and only if the Brenier map is homothetic affine, with quantitative stability via the sliced Poincaré-Korn constant of the source measure.