A partial solution of the Pompeiu problem

Abstract

A nonempty bounded open subset D of ℝ n is said to have the Pompeiu property if and only if for every continuous complex-valued function f on ℝ n which does not vanish identically there is a rigid motion σ of ℝ n onto itself — taking D onto σ(D) — such that the integral of f over σ(D) is not zero. This article gives a partial solution of the Pompeiu problem, the problem of finding all sets D with the Pompeiu property. In the special case that D is the interior of a homeomorphic image of an(n−1)-dimensional sphere, the main result states that if D has a portion of an(n−1)-dimensional real analytic surface on its boundary, then either D has the Pompeiu property or any connected real analytic extension of the surface also lies on the boundary of D. Thus, for example, any such region D having a portion of a hyperplane as part of its boundary must have the Pompeiu property, since the entire hyperplane cannot lie in the boundary of the bounded set D.