RUI: Geometric Tomography

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Geometric tomography uses data concerning sections by planes and projections on planes of geometric objects to obtain information about these objects. The latter include general compact sets, but often they are convex bodies, polytopes, star-shaped bodies, or finite sets. One advantage of this setting is that it becomes more probable that inverse problems have a unique solution. Generally, the a priori knowledge that the unknown object is of uniform density can be exploited to retrieve more information than would otherwise be possible. This can lead to algorithms that are more effective when few measurements are available, and less sensitive to measurement errors or noise. Geometric tomography has links to functional analysis, convex geometry, Minkowski geometry, and combinatorics. The project will continue the development of geometric tomography. New directions include fundamental new classification theorems for binary operations between, and symmetrization operations on, compact convex sets or star bodies; development of new tools for dealing with intersections of convex bodies, including an attack on a conjectured stronger form of the Brunn-Minkowski inequality; formulating and proving restricted Brunn-Minkowski inequalities, where the sets involved are contained in some fixed region of Euclidean space or the unit sphere; new algorithms for reconstruction from noisy data, in particular, from section functions, with proof of convergence for origin-symmetric convex bodies; progress towards a solution of Hammer's X-ray problem in higher dimensions and a conjecture on spherical X-rays of convex bodies; extension of previous results on convex and star bodies to log concave and radial measures; various other projects, for example renewed attacks on the discrete Aleksandrov projection problem and new inequalities of the Loomis-Whitney type or their reverses. Also included is a program designed to stimulate undergraduate research.CAT scanners are machines that save lives daily. They take X-rays in a number of different directions, and synthesize the information to create an image of a two-dimensional section of part of the body. The mathematics behind this process is called computerized tomography. It is very successful, but not perfect; the reconstructed image is only approximate, and to get a better picture with the same procedure one has to take more X-rays, causing greater expense and likelihood of side effects. In geometric tomography, only homogeneous objects are allowed - the density of the object is the same everywhere inside it. An example from medicine would be a bone or a kidney. One can use this extra information to find better reconstruction procedures. The scope of geometric tomography is actually much wider. Any measurement involving sections of a homogeneous object by lines or planes or its shadows on lines or planes can be considered. Because of this, it has many links to other areas, both in mathematics (there is a large overlap with convex geometry, the geometry of shapes without holes or dents) and outside. For example, a new technique called local stereology depends on measurements of planar sections of biological tissue; each section passes through a fixed point, usually the nucleus of a cell, and the measurements can be made optically rather than physically. This project continues the development of several aspects of the mathematics of geometric tomography. Also included is a program designed to stimulate undergraduate research.