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

Abstract

We prove that a connected globally conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: its outer and inner metric space structures are equivalent. Moreover, we show that generic $\mathbb{K}$-analytic germs as well as generic affine algebraic sets in ${\mathbb{K}}^n$, where $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$, are globally conic singular sub-manifolds. Consequently, a generic $\mathbb{K}$-analytic germ or a generic algebraic subset of ${\mathbb{K}}^n$ is Lipschitz Normally Embedded.