Solution of the Pompeiu Problem (i)

  title={Solution of the Pompeiu Problem (i)},
  author={Genqian Liu},
Abstract. A nonempty bounded open set Ω ⊂ R is said to have the Pompeiu property if and only if the only continuous function f on R for which the integral of f over σ(Ω) is zero for all rigid motions σ of R is f ≡ 0. In this paper, a longstanding open problem, the Pompeiu problem (or equivalently, the Schiffer conjecture), is completely solved in R. More precisely, we prove that among bounded open sets of R, each of which has a connected Lipschitz boundary, only the disks fail to have the… CONTINUE READING