Stereographic compactification and affine bi-Lipschitz homeomorphisms

• Published in: Glasgow Mathematical Journal
• Main Authors: Grandjean, Vincent, Oliveira, Roger
• Format: Article in Journal/Newspaper
• Language: English
• Published: Cambridge University Press (CUP) 2024
• Subjects: North Pole; Valette
• Online Access: http://dx.doi.org/10.1017/s001708952400017x; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S001708952400017X

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.