Pompeiu problem

In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929,

as follows. Suppose f is a nonzero continuous function defined on a Euclidean space, and K is a simply connected Lipschitz domain, so that the integral of f vanishes on every congruent copy of K. Then the domain is a ball.

A special case is Schiffer's conjecture.

References

{{citation | first=Dimitrie | last=Pompeiu |title=Sur certains systèmes d'équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables |journal=Comptes Rendus de l'Académie des Sciences, Série I |volume=188 |year=1929 | pages=1138–1139 }}
{{citation|title=Topics in mathematical analysis | volume=3 |series=Series on analysis, applications and computation | first=Paolo | last=Ciatti | publisher=World Scientific | year=2008 | isbn=978-981-281-105-9 }}

[https://www.math.u-szeged.hu/~odor/pompeiu.htm Pompeiu problem at Department of Geometry, Bolyai Institute, University of Szeged, Hungary]
[https://encyclopediaofmath.org/wiki/Pompeiu_problem Pompeiu problem at SpringerLink encyclopaedia of mathematics]
[https://www.scilag.net/problem/G-180522.1 The Pompeiu problem],
[https://www.scilag.net/problem/P-180522.1 Schiffer's conjecture],