Domains with algebraic X-ray transform

Abstract

Koldobsky, Merkurjev and Yaskin proved in (Koldobsky in Adv Math 320:876-886, 2017) that given a convex body $K \subset {\mathbb {R}}^n, \ n$ is odd, with smooth boundary, such that the volume of the intersection $K \cap L$ of K with a hyperplane $L \subset {\mathbb {R}}^n$ (the sectional volume function) depends polynomially on the distance t of L to the origin, then the boundary of K is an ellipsoid. In even dimension, the sectional volume functions are never polynomials in t, nevertheless in the case of ellipsoids their squares are. We conjecture that the latter property fully characterizes ellipsoids and, disregarding the parity of the dimension, ellipsoids are the only convex bodies with smooth boundaries whose sectional volume functions are roots (of some power) of polynomials. In this article, we confirm this conjecture for planar domains, bounded by algebraic curves. A multidimensional version in terms of chords lengths, i.e., of X-ray transform of the characteristic function, is given. The result is motivated by Arnold’s conjecture on characterization of algebraically integrable bodies.