Convex tomography in dimension two and three

Abstract

In this thesis we investigate a problem which is part of Geometric Tomography. Geometric Tomography is a branch of Mathematics which deals with the determination of a convex body (or other geometric objects) in n from the measure of its sections, projections, or both. In particular, it is focused on finding conditions sufficient to establish the minimum number of point X-rays needed to determine uniquely a convex body in n. The interest for this particular mathematical subject has its roots in the studies related to Tomography. Nowadays it is rare that someone has never heard of CAT scan (Computed Assisted Tomography). This medical diagnostic technique, born in the late 70s, allows us to reconstruct the image of a three-dimensional object by a large number of projections at different directions. It is useful to emphasize that CAT is a direct application of a pure mathematical instrument known as ‘the Radon transform”. From the mathematical point of view, the question of when a convex body, a compact convex set with nonempty interior, can be reconstructed by means of its X-ray, arose from problems (published in 1963) posed by Hammer in 1961 during a Symposium on convexity: « Suppose there is a convex hole in an otherwise homogeneous solid and that X-ray pictures taken are so sharp that the “darkness” at each point determines the length of a chord along an X-ray line. (No diffusion please). How many pictures must be taken to permit exact reconstruction of the body if: (a) The X-rays issue from a finite point source? (b) The X-rays are assumed parallel? » A convex body is a convex compact set with nonempty interior. We distinguish between two problems, according to the X-rays are at a finite point i ii source or at infinity. We are searching which properties a set of directions U must fulfill, in order to determine uniquely a convex body K by means of its (either parallel or point) X-rays, in the directions of U. When U is an infinite set then this reconstruction is possible and this follows from general theorems regarding the inversion of the Radon transform. This thesis consists of two parts. The first (Chapter 3) is concerned with the determination of a planar convex body from its i-chord functions, while the second part (Chapter 4) generalizes the planar results to the three-dimensional case. The main results provide a partial answer to the problem posed by R. J. Gardner: “How many point X-rays are needed to determine a convex body in n?” We use two analytic tools both considered in geometric tomography for the first time by K. J. Falconer. One is the i-chord function, which is related through the Funk theorem to the measures of the i-dimensional sections, when i is a positive integer. The other tool, inspired by a paper by D. C. Solmon, suggested the introduction of a kind of “Cavalieri Principle” for point X-rays, and has been later on extended to other dimensions and real values of i. The i-chord functions allow to translate in analytical form the information given by the ith section functions. Therefore, from an analytical point of view we have the following problem: “Suppose that K ! n is a convex body and let ph be some noncollinear points (some of them are possibly at infinity). Suppose, moreover, that we are given the i-chord functions at the points ph, with i " . Is K then uniquely determined among all convex bodies?” The i-chord functions !i,K can be seen as a generalization of the radial function of the convex body K. The latter is the function that gives the signed distance from the origin to the boundary of K. For integer values of the parameter i, the i-chord function is closely linked to the ith section function of a convex body, that is the function assigning to each subspace of dimension i the i-dimensional measure. When i = 1, the 1st section function coincides with the 1-chord function, that is the point X-ray of the convex body at the origin. In Chapter 3 we consider two planar problems. One problem consists of the determination of a planar convex body K from the i-chord functions, for i > 0, at two points when the line l passing through p1 and p2 meets the interior of K and the two points p1 and p2 are both exterior or interior to K. If the line l supports K, then the results hold for i # 1. The second result concerns the determination of a planar iii convex body K from its i-chord functions at three noncollinear points for 0 < i < 2. Chapter 4 deals with the problem of determining a three-dimensional convex body K from the i-chord functions at three noncollinear points non belonging