Draft:Lipschitz Regularity Theorem

The Lipschitz Regularity Theorem is a result in complex algebraic geometry that characterizes the complex analytic sets that are smooth from the metric point of view. It was proved by José Edson Sampaio, a Brazilian mathematician and professor at the Universidade Federal do Ceará.

The theorem states that any complex analytic set X in $$\mathbb C^n$$ that is Lipschitz regular at p must be smooth at p. In other words, if X is a complex analytic set in $$\mathbb C^n$$such that there exist open U of $$\mathbb C^n$$ that contains p and a bi-Lipschitz homeomorphism h: X $$\cap$$ U $$\to$$ B, then X is smooth at p, where B is an open ball of some Euclidean space.