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.
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