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

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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.