<p>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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Z_2(\mu ,\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to Wasserstein distance <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$W_2(\mu ,\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu ,\nu \in \mathcal {P}_2(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> <mo>∈</mo> <msub> <mi mathvariant="script">P</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are centred probabilities with prescribed variances.</p>

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

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2026年7月30日

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Karol Bołbotowski,
Guy Bouchitté

摘要

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 $d \ge 1$ $d \geq 1$ with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance $Z_2(\mu ,\nu )$ $Z_{2} (μ, ν)$ to Wasserstein distance $W_2(\mu ,\nu )$ $W_{2} (μ, ν)$ when $\mu ,\nu \in \mathcal {P}_2(\mathbb {R}^d)$ $μ, ν \in P_{2} (R^{d})$ are centred probabilities with prescribed variances.

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

摘要