Domains with radical-polynomial X-ray transform

TL;DR

This paper characterizes ellipsoids among smooth algebraic convex bodies by analyzing the behavior of their X-ray transform under small translations, revealing a unique geometric property related to polynomial roots.

Contribution

It establishes a novel characterization of ellipsoids based on the polynomial-root behavior of the X-ray transform under line translations.

Findings

If the X-ray transform behaves as the m-th root of a polynomial under small translations, then the boundary is an ellipsoid.

The result applies to bodies with smooth algebraic hypersurface boundaries.

Provides a new geometric criterion for identifying ellipsoids.

Abstract

Let $K$ be a compact convex body in $\mathbb R^n.$ For any affine line $L,$ denote $\widehat{\chi}_K(L)=\int_{L}\chi_K(x)dl(x),$ 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 $\widehat{\chi}_K(L)$ 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.