  A path-connected subanalytic subset in $\mathbb{R}^n$ is naturally equipped
with two metrics: the inner and the outer metrics. We say that a subset is
Lipschitz normally embedded (LNE) if these two metrics are equivalent. In this
article, we give some criteria for a subanalytic set to be LNE. It is a
fundamental question to know if the LNE property is conical, i.e., if it is
possible to describe the LNE property of a germ of a subanalytic set in terms
of the properties of its link. We answer this question by introducing a new
notion called link Lipschitz normally embedding. We prove that this notion is
equivalent to the LNE notion in the case of sets with connected links.
