Isometric rigidity of Wasserstein spaces over Euclidean spheres
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914942147362816 |
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| author | Gehér, György Pál Hrušková, Aranka Titkos, Tamás Virosztek, Dániel |
| author_facet | Gehér, György Pál Hrušková, Aranka Titkos, Tamás Virosztek, Dániel |
| contents | We study the structure of isometries of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ over the sphere endowed with the distance inherited from the norm of $\mathbb{R}^{n+1}$. We prove that $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ is isometrically rigid, meaning that its isometry group is isomorphic to that of $\left(\mathbb{S}^n,\|\cdot\|\right)$. This is in striking contrast to the non-rigidity of its ambient space $\mathcal{W}_2\left(\mathbb{R}^n,\|\cdot\|\right)$ but in line with the rigidity of the geodesic space $\mathcal{W}_2\left(\mathbb{S}^n,\sphericalangle\right)$. One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$. A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of $\mathcal{W}_p\left(\mathbb{S}^1, \|\cdot\|\right)$ for $1 \leq p<2.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05065 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Isometric rigidity of Wasserstein spaces over Euclidean spheres Gehér, György Pál Hrušková, Aranka Titkos, Tamás Virosztek, Dániel Metric Geometry Mathematical Physics Functional Analysis 54E40, 46E27 (Primary) 60A10, 60B05 (Secondary) We study the structure of isometries of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ over the sphere endowed with the distance inherited from the norm of $\mathbb{R}^{n+1}$. We prove that $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ is isometrically rigid, meaning that its isometry group is isomorphic to that of $\left(\mathbb{S}^n,\|\cdot\|\right)$. This is in striking contrast to the non-rigidity of its ambient space $\mathcal{W}_2\left(\mathbb{R}^n,\|\cdot\|\right)$ but in line with the rigidity of the geodesic space $\mathcal{W}_2\left(\mathbb{S}^n,\sphericalangle\right)$. One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$. A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of $\mathcal{W}_p\left(\mathbb{S}^1, \|\cdot\|\right)$ for $1 \leq p<2.$ |
| title | Isometric rigidity of Wasserstein spaces over Euclidean spheres |
| topic | Metric Geometry Mathematical Physics Functional Analysis 54E40, 46E27 (Primary) 60A10, 60B05 (Secondary) |
| url | https://arxiv.org/abs/2308.05065 |