Functions that are the Directed X-Ray of a Planar Convex Body

Abstract

We characterize functions that are the directed X-ray of a planar convex body from a source that is a positive distance from the body. In addition to a concavity condition the necessary and sufficient conditions involve the structure of points of zero curvature and a priori estimates for derivatives of the directed X-ray near supporting rays and points of zero curvature. The techniques employed also lead to explicit methods for constructing families of planar convex bodies with a common directed X-ray.