Domains with radical-polynomial X-ray transform

Abstract

Let $K$ be a compact convex body in ${\mathbb{R}}^n.$ For any affine line $L,$ denote ${\hat{\chi }}_K(L)={\int }_L{\chi }_K(x)dl(x),$ where $dl$ is the arc length measure, the $X$-ray transform of the characteristic function ${\chi }_K,$ i.e., the length of the chord $K\cap L.$ We prove that if $K$ is bounded by a $C^{\infty }$ real algebraic hypersurface $\partial K$ and the $X$-ray transform ${\hat{\chi }}_K(L)$ behaves, under small parallel translations of the line $L$ to the distance $t,$ as the $m$-th root of a polynomial of $t$, for some fixed $m\in \mathbb{N},$ then $\partial K$ is an ellipsoid.