Principal Investigator: Erwin Lutwak, Christoph Haberl, Deane Yang, Gaoyang Zhang
The aim of this project is to develop modern extensions of the classical Brunn-Minkowski theory (often called the theory of mixed volumes) in convex geometric analysis. Over the years, the PIs have established a number of fundamental affine isoperimetric inequalities and their analytic counterparts. Newly developed methods (by the PIs and others) will be exploited to attack long-conjectured inequalities. One of the PIs showed how an old notion of p-sum of convex bodies leads to an embryonic p-Brunn-Minkowski theory. The p-inequalities which arise in the new theory turn out to be almost invariably stronger than their classical counterparts. Recent advances have demonstrated the need to move to the next evolutionary step in the development of the Brunn-Minkowski theory: An Orlicz Brunn-Minkowski theory. Very recently, the PIs have succeeded in discovering the Orlicz analogue of two fundamental operators from the Brunn-Minkowski theory and have established the fundamental affine isoperimetric inequalities for them. The development of an Orlicz Brunn-Minkowski theory will be a main focus of the PIs efforts. Just as the classical Minkowski problem is a central focus of the classical Brunn-Minkowski theory, its p-analogue is central to the p-Brunn-Minkowski theory. The Orlicz versions of this elliptic partial differential equation are a central focus of the Orlicz Brunn-Minkowski theory and a sustained attack on this partial differential equation will be undertaken. The study of projections and intersections of convex bodies is fundamental and of significant practical value. The Brunn-Minkowski theory together with the cosine transform from harmonic analysis are ideal tools for the study of projections. There is a fascinating duality in convex geometry. One of the PIs initiated the study of the dual Brunn-Minkowski theory, which together with Radon transforms are the necessary tools for the study of intersections and to solve long-standing open problems as shown by previous work of one of the PIs and others. Continued development of the dual Brunn-Minkowski theory is proposed. Work of the PIs and others, indicates that there are interesting connections between Information Theory (a subject usually associated with Electrical Engineering) and convex geometric analysis. The PIs will continue their exploration of the interactions between these subjects.
The Brunn-Minkowski theory is the core of convex geometric analysis and is the foundation of subjects such as geometric tomography and stereology. These subjects investigate mathematical problems with strong practical backgrounds such as measuring the diffusion capacity of the lung, determining the volume of minerals in rocks, and detecting tumors in the human body. Numerous applications of the Brunn-Minkowski theory have been discovered in mathematics, science and engineering. For example, the reconstruction of hidden objects from information regarding sections (think CAT scan machines) involves a tomographic analysis. The proposed work should result in the development of new theories and techniques in convex geometric analysis which are potential new mathematical tools with applications to science and engineering.
The PIs continued their ongoing research, which has been funded by the NSF for the past 27 years, on convex geometric analysis. In particular, the PIs studied measures and structures of geometric objects such as volume, curvature, projections, and intersections. When the geometric objects studied are 3-dimensional models of real objects, convex geometric analysis provides the mathematical foundation for new powerful tools to analyze such objects. The PIs work also applies to abstract higher dimensional shapes that arise naturally in various subjects such as optimization, probability and statistics. A focus of PIs' research is centered around higher dimensional isoperimetric problems which originated from the classical isoperimetric inequality traced back to the Greeks and played a crucial role in the development of geometry. What is quite surprising is how important these problems are to many areas of mathematics and even other fields. It is for this reason that such higher dimensional isoperimetric problems have been the ongoing focus of many research mathematicians' investigations, including that of the PIs. In particular, the PIs discovered and continue to investigate actively subtle but strong connections between geometry and information theory. For example, certain variations of isoperimetric problems are related to questions on optimizing data transmission. Another expected outcome of the PIs' research efforts concerns the development of improved methods for identifying the shape and/or various parameters associated with geometric objects when only a limited amount of lower-dimensional geometric information is available. Such measurement problems are prevalent in science, engineering, and medicine. They are closely connected with the study of PIs on projection and intersection of geometric objects. They arise, for example, when the only data available is from probing an object by shooting narrowly focused rays of light, sound, or other types of radiation through the object. Or when one knows only the areas or perimeters of shadows of the object. Challenges like this form the basis of the subject known as geometric tomography. The mathematical ideas and tools developed by the PIs have laid (and are expected to continue to lay) some of the groundwork for advances in geometric tomography.
