Isometric embedding of the n-point spaces into the space of spaces for $n \leq 4$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910346885726208 |
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| author | Capdeville, Benjamin |
| author_facet | Capdeville, Benjamin |
| contents | In [The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces. Memoirs of the American Mathematical Society. American Mathematical Society, 2023], Sturm studied the space of all metric measure spaces up to isomorphism which he called The space of spaces. He also introduced for a natural number n the space of all n-points metric spaces. The aim of this article is to study if the embedding of this space in the space of spaces is isometric. Using results from [Haggai Maron and Yaron Lipman. (probably) concave graph matching. Advances in Neural Information Processing Systems, 31, 2018] and [Hiroshi Maehara. Euclidean embeddings of finite metric spaces. Discrete Mathematics, 2013], we prove that it is the case for $n \leq 4$ and for Euclidean metric spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_18156 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isometric embedding of the n-point spaces into the space of spaces for $n \leq 4$ Capdeville, Benjamin Metric Geometry In [The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces. Memoirs of the American Mathematical Society. American Mathematical Society, 2023], Sturm studied the space of all metric measure spaces up to isomorphism which he called The space of spaces. He also introduced for a natural number n the space of all n-points metric spaces. The aim of this article is to study if the embedding of this space in the space of spaces is isometric. Using results from [Haggai Maron and Yaron Lipman. (probably) concave graph matching. Advances in Neural Information Processing Systems, 31, 2018] and [Hiroshi Maehara. Euclidean embeddings of finite metric spaces. Discrete Mathematics, 2013], we prove that it is the case for $n \leq 4$ and for Euclidean metric spaces. |
| title | Isometric embedding of the n-point spaces into the space of spaces for $n \leq 4$ |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2402.18156 |