Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Critical values and level sets of distance functions in Riemannian,
  Alexandrov and Minkowski spaces   Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975)
proved that then, for almost all $r>0$, the level set (distance sphere,
$r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological
$(n-1)$-dimensional manifold. This result was improved by J.H.G. Fu (1985). We
show that Ferry's result is an easy consequence of the only fact that the
distance function $d(x)= \dist(x,F)$ is locally DC and has no stationary point
in $\R^n\setminus F$. Using this observation, we show that Ferry's (and even
Fu's) result extends to sufficiently smooth normed linear spaces $X$ with $\dim
X \in \{2,3\}$ (e.g., to $\ell^p_n, n=2,3, p\geq 2$), which improves and
generalizes a result of R. Gariepy and W.D. Pepe (1972). By the same method we
also generalize Fu's result to Riemannian manifolds and improve a result of K.
Shiohama and M. Tanaka (1996) on distance spheres in Alexandrov spaces.

21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci