Symmetry from sectional integrals for convex domains

Abstract

Let Ω be a bounded convex domain in Rn (n≥2). In this work, we prove that if there exists an integrable function f such that it's Radon transform over (n−1)-dimensional hyperplanes intersecting the domain Ω is a function G depending on the distance to the nearest parallel supporting hyperplane to Ω, then Ω is a ball and f is radial depending on certain assumptions on G. As a consequence we show that constants are not in range of Radon transform of integrable functions in dimension n≥3.