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  "body": "\n## Problem\n\nLet $n \\ge 2$ and let $K \\subseteq \\R^n$ be a bounded set with nonempty interior.\nThe set $K$ has the *Pompeiu property* if the only continuous function $f \\colon \\R^n \\to \\C$ with\n$$\\int_{\\sigma(K)} f \\, dx = 0 \\quad \\text{for every rigid motion } \\sigma \\text{ of } \\R^n$$\nis $f = 0$.\nHere a rigid motion is a composition of a rotation and a translation, and the integral is against [[Lebesgue measure|2Z2QVWZ262]].\nDetermine which sets have the Pompeiu property; in particular, decide whether every bounded simply connected Lipschitz domain without the Pompeiu property is a ball.\n\n## Remarks\n\nA ball does not have the Pompeiu property.\nFor the unit ball in $\\R^n$ and a suitable nonzero vector $a$, the function $f(x) = \\sin(a \\cdot x)$ has integral $0$ over every translate of the ball, and rotations map balls to balls.\nSquares and other polygons are known to have the Pompeiu property.\nBy a theorem of Williams (1976), for smooth contractible domains the conjectured answer is equivalent to [[Schiffer's conjecture|J05VKG4XQB]] on an overdetermined Neumann eigenvalue problem.\n\n## Explanations\n\n### L1 human\n\nSlide and rotate a fixed shape all over the plane, and suppose a function averages to zero on every copy of the shape.\nMust the function be zero everywhere?\nFor a square the answer is yes.\nFor a disc the answer is no: a suitable wave $\\sin(a x)$ averages to zero on every disc of the given radius.\nPompeiu's problem asks whether the disc is the only shape for which this happens.\n",
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      "first_stated": "1929",
      "posed_by": "Dimitrie Pompeiu"
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