An analogue of polynomially integrable bodies in even-dimensional spaces

A bounded domain K⊂Rⁿ is called polynomially integrable if the (n−1)-dimensional volume of the intersection K with a hyperplane Π polynomially depends on the distance from Π to the origin. It was proved in [7] 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.