Stereographic compactification and affine bi-Lipschitz homeomorphisms ...
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...
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| Format: | Text |
| Language: | unknown |
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arXiv
2023
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| Online Access: | https://dx.doi.org/10.48550/arxiv.2305.07469 https://arxiv.org/abs/2305.07469 |
| Summary: | 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. ... : 15 pages ... |
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