1 results
Stereographic compactification and affine bi-Lipschitz homeomorphisms
- Part of
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- Journal:
- Glasgow Mathematical Journal / Volume 66 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 16 May 2024, pp. 582-596
- Print publication:
- September 2024
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- Article
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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.
