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.