The Pompeiu problem

TL;DR

This paper discusses the Pompeiu problem, exploring conditions under which a domain must be a ball if certain integral conditions hold for all rigid motions, and provides new proofs for existing results.

Contribution

It introduces new short proofs for known results related to the Pompeiu problem and discusses related conjectures and their equivalences.

Findings

Complement of the domain is connected and path connected.

If the integral condition holds and the function is non-zero, the domain is conjectured to be a ball.

Several new short proofs of earlier results are provided.

Abstract

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.