A note on k-plane integral transforms

AbstractLet Π be a k-dimensional subspace of Rn, n ⩾ 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π⊥. The k-plane transform of a measurable function ƒ in the direction Π at the point x″ is defined by Lƒ(Π, x″) = ∝Πƒ(x′, x″) dx′. In this article certain a priori inequalities are established which show in particular that if ƒ ϵ Lp(Rn), 1 ⩽ p $̌nk, then ƒ is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces Lp(Rn), p ⩽ 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rn are also obtained

Planar Convex Bodies with a Common Directed X-ray

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

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

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