One point compactification and Lipschitz normally embedded definable subsets

Abstract

A closed subset of $\mathbb{R}^q$, definable in some given o-minimal structure, is Lipschitz normally embedded in $\mathbb{R}^q$ if and only if its one-point compactification is Lipschitz normally embedded in the unit sphere ${\bf S}^q$($ = \mathbb{R}^q \cup \{\infty \}$), i.e. the closure of its image by the inverse of the stereographic projection is Lipschitz normally embedded in ${\bf S}^q$. This implies that any closed connected unbounded definable subset of an Euclidean space is definably inner bi-Lipschitz homeomorphic to a Lipschitz normally embedded definable set.