Convex tomography and the characterisation of ellipsoids

This thesis explores some aspects of convex tomography. We look in some detail at formulations of problems in tomography in which information is known about chords or sections supporting convex bodies. Chapter 1 extends a result of Falconer on Hammer's X-Ray problem. Suppose that K and M are planar convex bodies containing, in a sense to be defined, sufficiently distinct convex bodies L1, L2 in intK [intersection] intM. Suppose further that whenever l is a line supporting L1 or L2 the chord lengths K [intersection] l and M [intersection] l agree. We prove that K = M. Chapter 2 applies the result obtained in Chapter 1 to prove a new characterisation of the ellipsoid. Suppose K, L are convex bodies in R3 with L [subset] intK, and that every section of K supporting L is centrally symmetric. We show that under certain conditions it is possible to prove that K is an ellipsoid. Chapter 3 explores some aspects of a problem which we refer to as the one body problem. Here we are concerned again with chord-lengths of planar convex bodies. Suppose that K is a convex body containing the convex body L in its interior. Further suppose that the lengths of the chords of K supporting L are given. We ask how much can be deduced about K. Several results are presented. In the final chapter. Chapter 4, we attempt to extend a result of Montejano. If every pair of sections of a convex body K through a point p are homothetic, is it true that K is a Euclidean ball. Montejano gave a positive result for the case p [episilon] intK. We provide a counterexample for the case p [epsilon] [delta]K and a much restricted result for the case p [set theory] K