Harmonic Analysis and Affine Isoperimetric Inequalities in Convex Geometry

This project focuses on two types of mathematical problems: tomographic problems and isoperimetric problems. Tomography concerns the retrieval of information about a geometric object based on limited information arising from its cross-sections or shadows (projections). For example, think of trying to determine the volume of a mountain based on the size of its shadows at different times of the day. Isoperimetric problems arise in geometry and optimization and have been of interest for thousands of years, dating back to ancient Greece. In a modern context, isoperimetric problems can involve studying the regularity of high-dimensional information. In complex data sets, high dimensionality often results in a regularizing effect. Tomographic and isoperimetric problems have been highly influential in many scientific disciplines, including physics, engineering, and computer science. Beyond their historical significance and applicability, these problems appeal to a wide audience because they are often intuitively stated and explained, while their solutions are difficult and require sophisticated mathematical techniques. The principal investigator of this project seeks to address problems arising naturally from geometry and harmonic analysis by employing techniques involving the Fourier transform, Radon transform, and other tools from various mathematical fields. Among these proposed problems are new affine invariant estimates for mixed-Sobolev norms, estimates for Radon and Cosine transforms (each abstractly represents the cross-sections and projections of an object, respectively) and their connections to long-standing problems including the recently resolved Bourgain slicing problem and the Busemann-Petty problems, and the very illusive Petty’s isoperimetric conjecture and Schneider’s difference body conjecture. The principal investigator will also seek interactions between these problems and other mathematical disciplines, including calculus of variations, partial differential equations, and probability theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.