Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Volume and Surface Area of Parallel Sets   The r-parallel set to a set A in a Euclidean space consists of all points
with distance at most r from A. We clarify the relation between the volume and
the surface area of parallel sets and study the asymptotic behaviour of both
quantities as r tends to 0. We show, for instance, that in general, the
existence of a (suitably rescaled) limit of the surface area implies the
existence of the corresponding limit for the volume, known as the Minkowski
content. A full characterisation is obtained for the case of self-similar
fractal sets. Applications to stationary random sets are discussed as well, in
particular, to the trajectory of the Brownian motion.

1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess