Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integral geometry of pairs of hyperplanes or lines   Crofton's formula of integral geometry evaluates the total motion invariant
measure of the set of $k$-dimensional planes having nonempty intersection with
a given convex body. This note deals with motion invariant measures on sets of
pairs of hyperplanes or lines meeting a convex body. Particularly simple
results are obtained if, and only if, the given body is of constant width in
the first case, and of constant brightness in the second case.

30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th