Abstract
A short and self-contained presentation of the results known about the Pompeiu problem is given and result is obtained. In particular, it is proved that if D1 has Pompeiu's property (P-property) then D2 has it, provided D2 is sufficiently close to D1 in the following sense: meas (D12\\D12) is sufficiently small. Here D12 = D1 ∪ D2 D12 = D1 D2 .
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