Uniqueness Results in Geometric Tomography (15rit189)


Alexander Koldobsky (University of Missouri)

(Kent State University)

(University of Alberta)

(Kent State Univeristy)


The Banff International Research Station will host the "Uniqueness results in geometric tomography" workshop in Banff from August 16 to August 23, 2015.

The study of geometric properties of convex bodies based on information about sections or projections of these bodies belongs to the area of geometric tomography and has important applications to many areas of mathematics and science, in general. Of paramount importance are questions about unique determination of convex bodies from the size of their sections or projections. For many years the dominating tools for proving uniqueness were those involving spherical harmonics and direct geometric methods. In recent years, we have seen a rapid development of new methods, based on Fourier analysis, which allowed to solve many open problems in convex geometry. The general idea is to express geometric characteristics of a body in terms of the Fourier transform and then use methods of harmonic analysis to solve geometric problems. This approach has led to several results including Fourier analytic solutions of the Busemann-Petty and Shephard problems, characterizations of intersection and projection bodies, extremal sections and projections of certain classes of bodies. These developments are described in the books ``Fourier Analysis in Convex Geometry" by Koldobsky and ``The Interface between Convex Geometry and Harmonic Analysis" by Koldobsky and Yaskin. The most recent results include solutions of several longstanding uniqueness problems, and the discovery of stability in volume comparison problems and its connection to hyperplane inequalities.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).