In this article we study Figalli and Gigli’s formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. We carry over classical characterisations of optimal plans to this setting and prove that the resulting spaces of measures, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {M}_p(X,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, are complete, separable and geodesic whenever the underlying space, X, is so. We also prove that, for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {M}_p(X,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> preserves the property of being non-branching, and for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> it preserves non-negative curvature in the Alexandrov sense. Finally, we prove isometric embeddings of generalised spaces of persistence diagrams <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {D}_p(X,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">D</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into the corresponding spaces <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {M}_p(X,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, generalising a result by Divol and Lacombe. As an application of this framework, we show that several known geometric properties of spaces of persistence diagrams follow from those of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {M}_p(X,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, including the fact that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {D}_2(X,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">D</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an Alexandrov space of non-negative curvature whenever X is a proper non-negatively curved Alexandrov space.

Optimal Partial Transport for Metric Pairs

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2026年7月24日

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Mauricio Che

摘要

In this article we study Figalli and Gigli’s formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. We carry over classical characterisations of optimal plans to this setting and prove that the resulting spaces of measures, $\mathcal {M}_p(X,A)$ $M_{p} (X, A)$ , are complete, separable and geodesic whenever the underlying space, X, is so. We also prove that, for $p>1$ $p > 1$ , $\mathcal {M}_p(X,A)$ $M_{p} (X, A)$ preserves the property of being non-branching, and for $p=2$ $p = 2$ it preserves non-negative curvature in the Alexandrov sense. Finally, we prove isometric embeddings of generalised spaces of persistence diagrams $\mathcal {D}_p(X,A)$ $D_{p} (X, A)$ into the corresponding spaces $\mathcal {M}_p(X,A)$ $M_{p} (X, A)$ , generalising a result by Divol and Lacombe. As an application of this framework, we show that several known geometric properties of spaces of persistence diagrams follow from those of $\mathcal {M}_p(X,A)$ $M_{p} (X, A)$ , including the fact that $\mathcal {D}_2(X,A)$ $D_{2} (X, A)$ is an Alexandrov space of non-negative curvature whenever X is a proper non-negatively curved Alexandrov space.