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

TL;DR

This paper establishes sharp inequalities between Zolotarev and Wasserstein distances in multidimensional probability spaces, extending previous results with optimal constants using a new duality principle for the Hessian.

Contribution

It introduces a new Kantorovich-Rubinstein duality for the Hessian and provides optimal bounds relating Zolotarev and Wasserstein distances in higher dimensions.

Findings

Extended Rio inequality to all dimensions with optimal constants.

Derived an optimal upper bound for the ratio of Zolotarev to Wasserstein distances.

Established sharp inequalities between these distances in $ ext{P}_2( eal^d)$.

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 $d \ge 1$ 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.