Theorem 1.If 8dffbb43e" title="Click to view the MathML source">Dhas P-property, then 8dffbb43e" title="Click to view the MathML source">Dis a ball.
Four equivalent formulations of the Pompeiu problem are discussed.
A domain 8dffbb43e" title="Click to view the MathML source">D has P-property if there exists an 982e3b1811bbf40f7e17da" title="Click to view the MathML source">f≠0, such that bbae53155cf120a1953484bdad219" title="Click to view the MathML source">∫Df(gx+y)dx=0 for all y∈R3 and all g∈SO(2), where 884a7f1b29a6675da438a2290d3e98" title="Click to view the MathML source">SO(2) is the rotation group.
The result obtained concerning the related symmetry problem is:
Theorem 2.If 8e95f9e764e0ec96deecfd6e21" title="Click to view the MathML source">(∇2+k2)u=0in 8dffbb43e" title="Click to view the MathML source">D, e5ec97c0ea404304bb" title="Click to view the MathML source">u∣S=1, uN∣S=0, and k>0is a constant, then 8dffbb43e" title="Click to view the MathML source">Dis a ball.