  We give a link criterion for normal embeddings of definable sets in o-minimal
structures. Namely, we prove that given a definable germ $(X, 0)\subset
(\mathbb{R}^n,0)$ with $(X\setminus\{0\},0)$ connected and a continuous
definable function $\rho: (X,0) \to \mathbb{R}_{\geq 0}$ such that $\rho(x)
\sim \|x\|$, then $(X,0)$ is Lipschitz normally embedded (LNE) if and only if
$(X,0)$ is link Lipschitz normally embedded (LLNE) with respect to $\rho$
(i.e., for $r>0$ small enough, $X\cap \rho^{-1}(r)$ is Lipschitz normally
embedded and its LNE constant is bounded by a constant $C$ independent of $r$).
This is a generalization of Mendes--Sampaio's result for the subanalytic case.
As an application, we give a counterexample to a question on the relation
between Lipschitz normal embedding and MD Homology asked by Bobadilla et al in
their paper about Moderately Discontinuous Homology.
