Tangent Cones And Regularity Of Real Hypersurfaces

We characterize C-1 embedded hypersurfaces of R-n as the only locally 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 C-1. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface X subset of R-n is C-1. Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a C-1 hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane. Finally we show that the last property is a special feature of real algebraic sets, in the sense that it does not hold in the real analytic category.