TANGENT CONES AND REGULARITY OF REAL HYPERSURFACES

Abstract. We characterize C1 embedded hypersurfaces of Rn as the only lo-cally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m < 3/2. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is C1. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particu-lar, any convex real analytic hypersurface X ⊂ Rn is C1. Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a C1 hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane. 1