Award: DMS 1312181, Principal Investigator: Erwin Lutwak, Deane Yang, Gaoyang Zhang
This proposal aims to develop both extensions and duals of the classical Brunn-Minkowski theory. One of the PIs has shown that for each p greater than 1 there is an Lp Brunn-Minkowski theory. Thus far, most work on this new theory has been limited to p greater than 1, but the PIs together with Boroczky have made progress in the singular case of p = 0. Just as the classical Minkowski problem is a central focus of the classical Brunn-Minkowski theory, its Lp analogue is central to the new theory. For p greater than 1, there has been significant progress on the elliptic PDE now known as the Lp Minkowski problem. A complete solution of the zero-th Minkowski problem would have a profound impact on a number of important questions. The PIs in collaboration with Boroczky have established both necessary and sufficient conditions for the existence of solutions to this logarithmic Minkowski problem when the prescribed "data"' is even. They have also established uniqueness in two dimensions. Proving uniqueness in higher dimensions is part of the proposed work. One particular focus of the PIs' research is the development of affine isoperimetric inequalities. Over the years the PIs have established a number of sharp affine isoperimetric inequalities and their analytic counterparts. New methods recently developed by the PIs will be used to extend earlier work. While the Brunn-Minkowski theory has been an effective tool for solving a variety of basic inverse problems involving data about projections of convex bodies onto subspaces, the study of a dual Brunn-Minkowski theory initiated by one of the PIs is ideal for dual questions involving intersections of convex bodies with subspaces. The PIs have recently discovered new dual curvature measures that arise naturally within the dual theory and believe that newly developed tools within the dual theory can be used to solve the PDE that is the natural dual analogue of the Christoffel-Minkowski problem. An intensive effort to solve this PDE is a central part of the work proposed. Previous work of the PIs has indicated fascinating parallels between information theory and both the Lp Brunn-Minkowski theory and its dual. The PIs will continue to investigate connections between a subject that is associated with Electrical Engineering and one that is considered pure mathematics.
The Brunn-Minkowski theory and its dual are the core of convex geometric analysis and are the foundation of Geometric Tomography. Geometric Tomography aims at retrieving information about a geometric object from data about its lower dimensional sections or projections. It has had and clearly will continue to have practical applications in science, engineering, and even medicine (think CAT scan machines). While isoperimetric inequalities go back to the ancient Greeks, many of the newer ones are applicable in geometry, analysis, and even engineering. The new inequalities and theories that the PIs propose to establish (and extend) should result in the development of mathematical tools that offer potential new applications to mathematics, science, and engineering.
