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Pompeiu Conjecture

A bounded domain Omega subset R^n has the Pompeiu property if every continuous function f:R^n->C for which

int_(sigma(Omega))f(x)dx=0

for every rigid motion sigma is identically zero. The Pompeiu conjecture asserted that a bounded domain with Lipschitz boundary homeomorphic to the (n-1)-sphere can fail to have the Pompeiu property only when it is a ball.

Guo (2026) gave computer-assisted constructions of bounded convex domains that are not balls and have real-analytic spherical boundaries. They fail the Pompeiu property in dimensions 3, 4, 6, 8, 10, and 14. For each construction, the Fourier transform of the indicator function of the domain vanishes on the unit sphere. The existence proofs use interval arithmetic for finite blocks and analytic bounds for the infinite tails.

See also

Fourier Transform, Indicator Function, Rigid Motion, Schiffer Conjecture

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References

Guo, J. "Convex Counterexamples to the Schiffer and Pompeiu Conjectures in Dimensions Three, Four, Six, Eight, Ten and Fourteen." 28 Sep 2026. https://arxiv.org/abs/2609.35419.

Cite this as:

Weisstein, Eric W. "Pompeiu Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PompeiuConjecture.html

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