Harmonic Analysis in Convex Geometry

Kent State University, Kent OH

Investigators

Abstract

The problems addressed in this project ask about properties of shapes in an ambient space (like a human heart in the body) that can be inferred from information about their shadows or slices (as in medical imaging). The advantage of the problems is that the solutions to many of them are "intuitively clear" not only to graduate students, but also to undergraduate students, and in some cases, even to high-school pupils! On the other hand, the answers are (very often) counterintuitive, requiring the use of the most advanced and sophisticated tools belonging to the different branches of modern mathematics. Moreover, many of the problems originated not in "pure math," but in medical imaging and tomography. For this reason the solutions might find very interesting biomedical applications. The current project is a continuation of the long-time collaboration between the two principal investigators. They will continue to use and develop the methods of harmonic analysis to solve problems arising in convex and discrete geometry. These problems include ones about Brunn-Minkowski-type inequalities for general measures, as well as questions related to different versions of slicing inequalities, including a slicing inequality for general measures and a discrete version of the slicing inequality. The principal investigators also plan to continue their work on the unique determination of convex bodies given information on the size (or some other properties) of projections and sections. This will involve a mixture of topological, probabilistic, and Fourier-analytic methods, and part of the work will be concentrated around a classical problem of Bonnensen that the principal investigators and their collaborators have previously solved in even dimensions. Many of the techniques developed in that work are new, and there are high hopes that the methods could help to solve a number of classical (but still open) questions in geometric tomography.