# On polynomially integrable domains in Euclidean spaces

## TL;DR

This paper investigates whether odd-dimensional ellipsoids are uniquely characterized by the polynomial nature of their Radon transform in Euclidean spaces, exploring a partial verification of this property.

## Contribution

It provides partial results and discussion on the uniqueness of odd-dimensional ellipsoids based on the polynomial property of their Radon transform.

## Key findings

- Ellipsoids have algebraic Radon transforms.
- In odd dimensions, the Radon transform of ellipsoids is polynomial in t.
- The paper discusses whether these properties uniquely identify ellipsoids.

## Abstract

Let $D$ be a bounded domain in $\mathbb R^n,$ with smooth boundary. Denote $V_D(\omega,t), \ \omega \in S^{n-1}, t \in \mathbb R,$ the Radon transform of the characteristic function $\chi_{D}$ of the domain $D,$ i.e., the $(n-1)-$ dimensional volume of the intersection $D$ with the hyperplane $\{x \in \mathbb R^n: <\omega,x>=t \}.$ If the domain $D$ is an ellipsoid, then the function $V_D$ is algebraic and if, in addition, the dimension $n$ is odd, then $V(\omega,t)$ is a polynomial with respect to $t.$ Whether odd-dimensional ellipsoids are the only bounded smooth domains with such a property? The article is devoted to partial verification and discussion of this question.