Sharp inequalities between Zolotarev and Wasserstein distances in P₂(Rᵈ)

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 d ≥ 1 with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance Z₂(μ ,ν ) Z 2 ( μ , ν ) to Wasserstein distance W₂(μ ,ν ) W 2 ( μ , ν ) when μ ,ν ∈ P₂(Rᵈ) μ , ν ∈ P 2 ( R d ) are centred probabilities with prescribed variances.