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 ${\mathbf{S}}^q$($={\mathbb{R}}^q\cup \{\infty \}$), i.e. the closure of its image by the inverse of the stereographic projection is Lipschitz normally embedded in ${\mathbf{S}}^q$. 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.

Abstract is extended. Introduction is modified accordingly to the addition of two new sections. Old sections 2 to 7 are now 1 to 6. We have added applications: Section 7 about the existence of LNE model of closed unbounded definable sets; and Section 8 presenting examples (definable and not) and their respective LNE nature in different compactifications