Symmetry from sectional integrals for convex domains

Abstract

Let $\Omega$ be a bounded convex domain in ${\mathbb{R}}^n$ ($n\ge 2$). 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 $\Omega$ is a strictly positive function of distance to the nearest parallel supporting hyperplane to $\Omega$, then $\Omega$ is a ball and the function $f$ is a unique radial function about the centre of $\Omega$.