## Abstract

A bounded domain K⊂R^{n} 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.

Original language | English |
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Article number | 127071 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 529 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jan 2024 |

### Bibliographical note

Publisher Copyright:© 2023 Elsevier Inc.

## Keywords

- Ellipsoids
- Hilbert transform
- Polynomials
- Radon transform
- Volumes