Stereographic compactification and affine bi-Lipschitz homeomorphisms

Let σ q \,:\,Rq →Sq Nq be the inverse of the stereographic projection with center the north pole Nq . Let Wᵢ be a closed subset of Rqᵢ , for $i=1,2$ . Let Φ \,:\,W₁ → W₂ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism σ q₂∘ Φ ∘ σ q₁⁻¹ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at Nq₁ with value Nq₂ whenever W₁ 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.