Stereographic compactification and affine bi-Lipschitz homeomorphisms
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916495380971520 |
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| author | Grandjean, Vincent Oliveira, Roger |
| author_facet | Grandjean, Vincent Oliveira, Roger |
| contents | Let $σ_q : \mathbb{R}^q \to {\bf S}^q \setminus N_q$ be the inverse of the stereographic projection with centre the north pole $N_q$. Let $W_i$ be a closed subset of $\mathbb{R}^{q_i}$, for $i=1,2$. Let $Φ:W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $σ_{q_2}\circ Φ\circ σ_{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ is unbounded. As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: 1) Sampaio's tangent cone result; 2) Links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07469 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stereographic compactification and affine bi-Lipschitz homeomorphisms Grandjean, Vincent Oliveira, Roger Metric Geometry Logic Let $σ_q : \mathbb{R}^q \to {\bf S}^q \setminus N_q$ be the inverse of the stereographic projection with centre the north pole $N_q$. Let $W_i$ be a closed subset of $\mathbb{R}^{q_i}$, for $i=1,2$. Let $Φ:W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $σ_{q_2}\circ Φ\circ σ_{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ is unbounded. As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: 1) Sampaio's tangent cone result; 2) Links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette. |
| title | Stereographic compactification and affine bi-Lipschitz homeomorphisms |
| topic | Metric Geometry Logic |
| url | https://arxiv.org/abs/2305.07469 |