Tangent cones and regularity of real hypersurfaces

Abstract

Abstract We characterize 𝒞 1 $\mathcal {C}^1$ embedded hypersurfaces of 𝐑 n $\mathbf {R}^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m < 3 / 2 $m<3/2$ . It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is 𝒞 1 $\mathcal {C}^1$ . 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 ⊂ 𝐑 n $X\subset \mathbf {R}^n$ is 𝒞 1 $\mathcal {C}^1$ . Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a 𝒞 1 $\mathcal {C}^1$ 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.