Minkowski weak embedding theorem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917751862329344 |
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| author | Garitsis, Efstathios Konstantinos Chrontsios Troscheit, Sascha |
| author_facet | Garitsis, Efstathios Konstantinos Chrontsios Troscheit, Sascha |
| contents | A well-known theorem of Assouad states that metric spaces satisfying the doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean spaces. Due to the invariance of many geometric properties under bi-Lipschitz maps, this result greatly facilitates the study of such spaces. We prove a non-injective analog of this embedding theorem for spaces of finite Minkowski dimension. This allows for non-doubling spaces to be weakly embedded and studied in the usual Euclidean setting. Such spaces often arise in the context of random geometry and mathematical physics with the Brownian continuum tree and Liouville quantum gravity metrics being prominent examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minkowski weak embedding theorem Garitsis, Efstathios Konstantinos Chrontsios Troscheit, Sascha Metric Geometry Classical Analysis and ODEs Dynamical Systems A well-known theorem of Assouad states that metric spaces satisfying the doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean spaces. Due to the invariance of many geometric properties under bi-Lipschitz maps, this result greatly facilitates the study of such spaces. We prove a non-injective analog of this embedding theorem for spaces of finite Minkowski dimension. This allows for non-doubling spaces to be weakly embedded and studied in the usual Euclidean setting. Such spaces often arise in the context of random geometry and mathematical physics with the Brownian continuum tree and Liouville quantum gravity metrics being prominent examples. |
| title | Minkowski weak embedding theorem |
| topic | Metric Geometry Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2408.09063 |