An analog of polynomially integrable bodies in even-dimensional spaces

TL;DR

This paper explores a modified concept of polynomial integrability in even-dimensional spaces, proving that ellipsoids are uniquely characterized by this property among smooth convex bodies.

Contribution

It introduces a new notion of polynomial integrability involving the Hilbert transform and proves ellipsoids are the only convex smooth bodies satisfying this in even dimensions.

Findings

Ellipsoids are the unique convex smooth bodies with the modified polynomial integrability property in even dimensions.

The paper extends the classification of polynomially integrable bodies to a broader class involving the Hilbert transform.

No other convex smooth bodies besides ellipsoids satisfy the modified property in even-dimensional spaces.

Abstract

A bounded domain $K \subset \mathbb R^n$ 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.