An Ellipse Criterion for Exact Nonuniqueness in the Planar Interior Radon Problem

Abstract

We characterize exact nonuniqueness in the planar interior Radon problem when a bounded open convex inner domain is compactly contained in a bounded open convex outer domain, without assuming central symmetry. A nonzero smooth function compactly supported in the outer domain and having zero integrals over every line meeting the inner domain exists if and only if an ellipse with no prescribed center contains the closure of the inner domain and is compactly contained in the outer domain. The same criterion holds for $L^1$ functions with compactly contained essential support whose Radon transforms vanish almost everywhere on those lines. The smooth witness may be chosen real, centrally symmetric about the ellipse center, and strictly negative on an open neighborhood of the ellipse. Necessity combines an unoriented-line parametrization, a representing measure, a vector-valued Hardy model, and an operator-pencil argument that extracts the required ellipse from a $2 \times 2$ matrix. For concentric open squares, the sharp inner-to-outer half-side threshold is $1/\sqrt{2}$. This yields counterexamples to Conjecture 1.2 of Boman (2021) and to Theorem 40.1 of Boman (2025).