  The main result states that a connected conic singular sub-manifold of a
Riemannian manifold, compact when the ambient manifold is non-Euclidean, is
Lipschitz Normally Embedded: the outer and inner metric space structures are
metrically equivalent. We also show that a closed subset of $\mathbb{R}^n$ is a
conic singular sub-manifold if and only if its closure in the one point
compactification ${\bf S}^n =\mathbb{R}^n\cup \infty$ is a conic singular
sub-manifold. Consequently the connected components of generic affine real and
complex algebraic sets are conic at infinity, thus are Lipschitz Normally
Embedded.
