Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tangent cones and regularity of real hypersurfaces   We characterize embedded $\C^1$ 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
$X\subset\R^n$ 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.

13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph