Planar Convex Bodies with a Common Directed X-ray

Abstract

We study X(K), the set of convex bodies in the plane with the same directed X-ray as the convex body K. We show that X(K) is complete in the metrics of the uniform and Lp norms. In fact these metrics turn out to be equivalent even though X(K) is almost always infinite dimensional. In addition, we characterize the compact subsets of X(K) and determine necessary and sufficient conditions for X(K) to be uniformly bounded.