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 ${\mathbf{S}}^n={\mathbb{R}}^n\cup \mathrm{\infty }$ 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.