Isoperimetric Inequalities

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The Brunn-Minkowski theory, often called the theory of mixed volumes, is the very core of convex geometric analysis. Over the years numerous applications to---or connections with---other fields of mathematics, science, and engineering have been discovered, including partial differential equations, algebraic geometry, statistics, number theory, Minkowski and Finsler geometry, stereology, information theory. The goal of this proposal is the development of extensions and duals of the classical Brunn-Minkowski theory. A special focus is the development of affine isoperimetric inequalities. There have been numerous applications of affine inequalities in PDEs, Banach space geometry, geometric tomography, and even robot vision. Another special focus will be the development of the analytic counterparts of affine isoperimetric inequalities. New methods developed by the PIs will be used to attempt to make progress on a number of long-conjectured affine isoperimetric inequalities. Another main focus of the proposal concerns a prescribed curvature problem, known as the Lp Minkowski problem, that arises naturally within the extended Brunn-Minkowski theory. Work of the PIs indicates that there are interesting potential connections between information theory and convex geometric analysis. These connections will be exploited with the aim of establishing new geometric results inspired by their information theory counterparts.The classical Brunn-Minkowski theory has provided powerful tools needed to solve a variety of basic inverse problems where the only "data" available involves information regarding the projections of convex bodies onto lower-dimensional subspaces (such as lines and planes). However, the tools of the classical Brunn-Minkowski theory have been of little value in dealing with the dual of these questions, where "projections onto subspaces" are replaced by "intersections with subspaces". The PIs are continuing efforts to develop a dual Brunn-Minkowski theory. This dual theory has already led to the solution of a longstanding problem where the known information involved the intersections of the unknown bodies with planes. Such "inverse problems" are basic in not only mathematics but also science and engineering.