Harmonic Analysis in Convex Geometry

Kent State University, Kent OH

Investigators

Abstract

This project investigates questions in convex geometry using methods originating in harmonic analysis. The definition of convexity is elementary: a set is convex if it contains the straight-line path between any two of its elements. Yet the study of such sets leads to a rich and intricate branch of modern geometry, exhibiting unexpected connections to diverse areas of pure and applied mathematics, including probability, number theory, linear programming, computational geometry, and tomography. Significant parts of the project will rely on new techniques for constructing convex bodies and exposing their geometric properties using the Fourier transform, which represents mathematical objects via their frequency decomposition. The project will provide research opportunities for students and postdoctoral fellows. In addition, the principal investigators will continue their organization of seminars and other meetings including the Informal Analysis Seminar, a twice yearly meeting where researchers from around the world present lectures aimed at early career researchers. The project involves the study of questions in convex geometry using tools from harmonic analysis and differential geometry. Some of the proposed research questions arose in the context of earlier work of the principal investigators, where Fourier analytic approaches were used to study inequalities involving volumes and mixed volumes. These approaches have proved indispensable in the solution of longstanding problems in convex geometry and geometric tomography. Many natural questions in convex geometry involve relationships between volumes of pairs of convex sets; among these is an open question posed by K. Mahler in 1939. Another source of inspiration is the so-called `Scottish Book’, an informal collection of about 200 problems in mathematics which originated in the 1930s. Stanislav Ulam’s 19th Problem, taken from the Scottish Book, asks for a characterization of the convex sets of a fixed density (less than one) which float in an equilibrium position in every orientation (in water, of density one). Surprisingly, round balls are not the only such sets, due to a recent result of one of the principal investigators of this project. Part of the current project involves a further study of Ulam’s 19th Problem in relation to the properties of sections and projections of convex bodies. A second research direction involves the Mahler conjecture for volumes of three-dimensional convex bodies. This conjecture is known to hold under an additional symmetry assumption, and the project will consider the general version of this conjecture in three dimensions. The project will provide ample opportunities for participation in research by early-career researchers, including graduate students and postdocs. 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.