Isometric rigidity of the Wasserstein space over the plane with the maximum metric
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915390160896000 |
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| author | Balogh, Zoltán M. Kiss, Gergely Titkos, Tamás Virosztek, Dániel |
| author_facet | Balogh, Zoltán M. Kiss, Gergely Titkos, Tamás Virosztek, Dániel |
| contents | We study $p$-Wasserstein spaces over the branching spaces $\mathbb{R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb{R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_07051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isometric rigidity of the Wasserstein space over the plane with the maximum metric Balogh, Zoltán M. Kiss, Gergely Titkos, Tamás Virosztek, Dániel Metric Geometry Mathematical Physics Functional Analysis Primary: 54E40, 46E27. Secondary: 60B05 We study $p$-Wasserstein spaces over the branching spaces $\mathbb{R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb{R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid. |
| title | Isometric rigidity of the Wasserstein space over the plane with the maximum metric |
| topic | Metric Geometry Mathematical Physics Functional Analysis Primary: 54E40, 46E27. Secondary: 60B05 |
| url | https://arxiv.org/abs/2411.07051 |