The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911782140903424 |
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| author | Orzechowski, Kamil |
| author_facet | Orzechowski, Kamil |
| contents | We show some results related to the classical Banach-Tarski paradox in the setting of finite-dimensional normed spaces over a non-Archimedean valued field $K$. For instance, all balls and spheres in $K^n$, and the whole space $K^n$ (for $n\ge 2$) are paradoxical with respect to certain groups of isometries of $K^n$.
If $K$ is locally compact (e.g., $K$ is the field $\mathbb{Q}_p$ of $p$-adic numbers for any prime number $p$), any two bounded subsets of $K^n$ with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of $K^n$ (for $n\ge 2$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14772 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields Orzechowski, Kamil Functional Analysis Group Theory Representation Theory Primary 47S10, Secondary 46S10, 12J25, 26E30, 20E05, 20H20, 46B04, 05A18, 03E25 We show some results related to the classical Banach-Tarski paradox in the setting of finite-dimensional normed spaces over a non-Archimedean valued field $K$. For instance, all balls and spheres in $K^n$, and the whole space $K^n$ (for $n\ge 2$) are paradoxical with respect to certain groups of isometries of $K^n$. If $K$ is locally compact (e.g., $K$ is the field $\mathbb{Q}_p$ of $p$-adic numbers for any prime number $p$), any two bounded subsets of $K^n$ with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of $K^n$ (for $n\ge 2$). |
| title | The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields |
| topic | Functional Analysis Group Theory Representation Theory Primary 47S10, Secondary 46S10, 12J25, 26E30, 20E05, 20H20, 46B04, 05A18, 03E25 |
| url | https://arxiv.org/abs/2402.14772 |