Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Pompeiu problem   Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}'$, where $\mathcal{S}'$ is the
Schwartz class of distributions, and $$\int_{\sigma (D)} f(x) dx = 0 \quad
\forall \sigma \in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain,
the closure $\bar{D}$ of which is diffeomorphic to a closed ball. Then the
complement of $\bar{D}$ is connected and path connected. Here $G$ denotes the
group of all rigid motions in $\R^n$. This group consists of all translations
and rotations.
  It is conjectured that if $f\neq 0$ and (*) holds, then $D$ is a ball. Other
conjectures, equivalent to the above one, are formulated and discussed. Several
new short proofs are given for the earlier proved results.

11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph