Sharp inequalities between Zolotarev and Wasserstein distances in P₂(ℝ^d)

2025/10/31 by Karol Bołbotowski, Bołbotowski, Karol, Guy Bouchitté +1
Mathematics · #Geometric Analysis and Curvature Flows #Advanced Harmonic Analysis Research #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2511.00232

Abstract

Based on a new Kantorovich-Rubinstein duality principle for the Hessian that was recently established by the two authors, we extend the Rio inequality to any dimension d ≥ 1 with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance Z₂(μ,ν) to Wasserstein distance W₂(μ,ν) when μ,ν∈ P₂(ℝ^d) are centred probabilities with prescribed variances.

Citations

• Improved rates of convergence for the multivariate Central Limit Theorem in Wasserstein distance
• Higher-order Stein kernels for Gaussian approximation
• Fundamentals of Stein's method
• Polar factorization and monotone rearrangement of vector‐valued functions

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