For the uniform measure on any compact convex body with nonempty interior in ℝᵈ, d≥2, and targets supported in one fixed compact set, the L² distance between their quadratic optimal maps is at most C times the one-third power of the targets’ 2-Wasserstein distance.
For the uniform measure on any compact convex body with nonempty interior in ℝᵈ, d≥2, and targets supported in one fixed compact set, the L² distance between their quadratic optimal maps is at most C times the one-third power of the targets’ 2-Wasserstein distance. No larger uniform exponent works in general, even for three-atom targets on a fixed cube.
For the uniform measure on any compact convex body with nonempty interior in ℝᵈ, d≥2, and targets supported in one fixed compact set, the L² distance between their quadratic optimal maps is at most C times the one-third power of the targets’ 2-Wasserstein distance. No larger uniform exponent works in general, even for three-atom targets on a fixed cube.
If correct, determines the exact uniform stability exponent and refutes the proposed square-root exponent.
What is the optimal uniform Wasserstein stability exponent for Brenier maps from a convex uniform source?
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