Stereographic compactification and affine bi-Lipschitz homeomorphisms
Abstract Let $\sigma _q \,:\,{{\mathbb{R}}^q} \to{\textbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$ . Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$ , for $i=1,2$ . Let $\Phi \,:\,W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main r...
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crcambridgeupr:10.1017/s001708952400017x 2024-06-16T07:42:04+00:00 Stereographic compactification and affine bi-Lipschitz homeomorphisms Grandjean, Vincent Oliveira, Roger 2024 http://dx.doi.org/10.1017/s001708952400017x https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S001708952400017X en eng Cambridge University Press (CUP) https://www.cambridge.org/core/terms Glasgow Mathematical Journal page 1-15 ISSN 0017-0895 1469-509X journal-article 2024 crcambridgeupr https://doi.org/10.1017/s001708952400017x 2024-05-22T12:54:03Z Abstract Let $\sigma _q \,:\,{{\mathbb{R}}^q} \to{\textbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$ . Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$ , for $i=1,2$ . Let $\Phi \,:\,W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ 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 and (2) links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette. Article in Journal/Newspaper North Pole Cambridge University Press North Pole Valette ENVELOPE(-44.600,-44.600,-60.750,-60.750) Glasgow Mathematical Journal 1 15 |
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Open Polar |
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Cambridge University Press |
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crcambridgeupr |
language |
English |
description |
Abstract Let $\sigma _q \,:\,{{\mathbb{R}}^q} \to{\textbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$ . Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$ , for $i=1,2$ . Let $\Phi \,:\,W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ 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 and (2) links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette. |
format |
Article in Journal/Newspaper |
author |
Grandjean, Vincent Oliveira, Roger |
spellingShingle |
Grandjean, Vincent Oliveira, Roger Stereographic compactification and affine bi-Lipschitz homeomorphisms |
author_facet |
Grandjean, Vincent Oliveira, Roger |
author_sort |
Grandjean, Vincent |
title |
Stereographic compactification and affine bi-Lipschitz homeomorphisms |
title_short |
Stereographic compactification and affine bi-Lipschitz homeomorphisms |
title_full |
Stereographic compactification and affine bi-Lipschitz homeomorphisms |
title_fullStr |
Stereographic compactification and affine bi-Lipschitz homeomorphisms |
title_full_unstemmed |
Stereographic compactification and affine bi-Lipschitz homeomorphisms |
title_sort |
stereographic compactification and affine bi-lipschitz homeomorphisms |
publisher |
Cambridge University Press (CUP) |
publishDate |
2024 |
url |
http://dx.doi.org/10.1017/s001708952400017x https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S001708952400017X |
long_lat |
ENVELOPE(-44.600,-44.600,-60.750,-60.750) |
geographic |
North Pole Valette |
geographic_facet |
North Pole Valette |
genre |
North Pole |
genre_facet |
North Pole |
op_source |
Glasgow Mathematical Journal page 1-15 ISSN 0017-0895 1469-509X |
op_rights |
https://www.cambridge.org/core/terms |
op_doi |
https://doi.org/10.1017/s001708952400017x |
container_title |
Glasgow Mathematical Journal |
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1 |
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15 |
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1802009445925912576 |