Global Lipschitz geometry of conic singular sub-manifolds with applications to algebraic sets

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.