On Polynomially Integrable Domains in Euclidean Spaces

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 χ _D of the domain D, i.e., the (n − 1)-dimensional volume of the intersection D with the hyperplane $\{x \in \mathbb R^n: <\omega ,x>=t \}.$ 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 (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.

To the memory of Alexander Vasiliev