The Pompeiu problem
The longstanding open problem of integral geometry, the Pompeiu problem states that if the integral of a nonzero continuous function on the n dimensional Euclidean space vanish on every congruent copy of the domain K then the domain is a ball, provided K is ``nice'' - that is bounded and it admits connected Lipschitzian boundary. (Convex bodies obviously satisfies this condition.)
The problem was posed originally by D. Pompeiu, Sur certain systèmes d'équations linéaries et sur une propriété intégrale des fonktions de plusieurs variables, C. R. Acad. Sci. Paris, 188, (1929), 1138 - 1139. It has attracted a large interest and has several equivalent forms. The most important is Schiffer's conjecture.
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