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.