Theorem 1.If Dhas 6249511d5c96b9f1990863714afa" title="Click to view the MathML source">P-property, then Dis a ball.
Four equivalent formulations of the Pompeiu problem are discussed.
A domain D has 6249511d5c96b9f1990863714afa" title="Click to view the MathML source">P-property if there exists an f≠0, such that ∫Df(gx+y)dx=0 for all y∈R3 and all g∈SO(2), where SO(2) is the rotation group.
The result obtained concerning the related symmetry problem is:
Theorem 2.If (∇2+k2)u=0in D, u∣S=1, uN∣S=0, and k>0is a constant, then Dis a ball.