mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Pompeiu problem is a conjecture in integral geometry, named for

Dimitrie Pompeiu Dimitrie D. Pompeiu (; – 8 October 1954) was a Romanian mathematician, professor at the University of Bucharest, titular member of the Romanian Academy, and President of the Chamber of Deputies. Biography He was born in 1873 in Broscăuți, ...

, who posed the problem in 1929, as follows. Suppose ''f'' is a nonzero

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

defined on a Euclidean space, and ''K'' is a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

Lipschitz domain In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The ...

, so that the

integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

of ''f'' vanishes on every congruent copy of ''K''. Then the domain is a

ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

. A special case is Schiffer's conjecture.

References

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External links

Pompeiu problem at Department of Geometry, Bolyai Institute, University of Szeged, Hungary

Pompeiu problem at SpringerLink encyclopaedia of mathematics

The Pompeiu problem

Schiffer's conjecture
Mathematical analysis Integral geometry Conjectures Unsolved problems in geometry {{mathanalysis-stub