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  1. ; ; ; (, Journal of mathematical analysis and applications)
    A bounded domain $$K \subset R^n$$ is called polynomially integrable ifthe $(n-1)$-dimensional volume of the intersection $$K$$ with a hyperplane $$\Pi$$ polynomially depends on the distance from $$\Pi$$ to the origin. It was proved in \cite{KMY} that there are no such domains with smooth boundary if $$n$$ is even, and if $$n$$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $$n$$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property. 
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