A hypersurface containing the support of a Radon transform must be an ellipsoid. I

TL;DR

This paper proves that if the support of a Radon transform of a compactly supported distribution is contained in the tangent planes of a boundary, then that boundary must be an ellipsoid, providing a new proof of a related theorem.

Contribution

It establishes a geometric characterization of boundaries supporting Radon transforms, specifically showing they must be ellipsoids, and offers a new proof of a recent related theorem.

Findings

Support of Radon transform implies boundary is an ellipsoid

Provides a new proof of a theorem related to Radon transforms

Connects geometric properties with Radon transform support

Abstract

If the Radon transform of a compactly supported distribution $f \ne 0$ in $\mathbb R^n$ is supported on the set of tangent planes to the boundary $\partial D$ of a bounded convex domain $D$, then $\partial D$ must be an ellipsoid. As a corollary of this result we get a new proof of a recent theorem of Koldobsky, Merkurjev, and Yaskin, which settled a special case of a conjecture of Arnold that was motivated by a famous lemma of Newton.