Sharp inequalities between Zolotarev and Wasserstein distances in $\mathrm{P}_2(\mathbb{R}^d)$

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 $d \ge 1$ with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance $Z_2(\mu,\nu)$ to Wasserstein distance $W_2(\mu,\nu)$ when $\mu,\nu \in \mathrm{P}_2(\mathbb{R}^d)$ are centred probabilities with prescribed variances.