Domains with radical-polynomial X-ray transform

Let $K$ be a compact convex body in $\mathbb R^n.$ For any affine line $L,$ denote $\widehatχ_K(L)=\int_{L}χ_K(x)dl(x),$ where $dl$ is the arc length measure, the $X$-ray transform of the characteristic function $χ_K,$ 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 $\widehatχ_K(L)$ 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.

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