Pompeiu problem
Encyclopedia
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Pompeiu problem is a conjecture in integral geometry
Integral geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...
, named for Dimitrie Pompeiu
Dimitrie Pompeiu
-Biography:After studying in Dorohoi and Bucharest, he went to France, where he studied mathematics at the University of Paris . He obtained a Ph.D. degree in mathematics in 1905 with a thesis, On the continuity of complex variable functions, written under the direction of Henri Poincaré...
, who posed the problem in 1929,
as follows. Suppose f is a nonzero continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
defined on a Euclidean space, and K is a simply connected Lipschitz domain
Lipschitz domain
In mathematics, a Lipschitz domain 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...
, so that the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of f vanishes on every congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
copy of K. Then the domain is a ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
.
A special case is Schiffer's conjecture.