An analogue of polynomially integrable bodies in even-dimensional spaces

A bounded domain K subset of R-n is called polynomially integrable if the (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 [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.

Automated summary

This article modifies the concept of polynomial integrability for even dimensions and proves that ellipsoids are the only convex infinitely smooth bodies satisfying this property.