On polynomially integrable domains in Euclidean spaces

Abstract

Let $D$ be a bounded domain in $\mathbb R^n,$ with smooth boundary. Denote $V_D(ω,t), \ ω\in S^{n-1}, t \in \mathbb R,$ the Radon transform of the characteristic function $χ_{D}$ of the domain $D,$ i.e., the $(n-1)-$ dimensional volume of the intersection $D$ with the hyperplane $\{x \in \mathbb R^n: <ω,x>=t \}.$ If the domain $D$ is an ellipsoid, then the function $V_D$ is algebraic and if, in addition, the dimension $n$ is odd, then $V(ω,t)$ 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.