Isoperimetric Inequalities

Polytechnic University Of New York, Brooklyn NY

Investigators

Abstract

Abstract Award: DMS-0405707 Principal Investigator: Erwin Lutwak, Deane Yang, Gaoyong Zhang This project continues a long term program to develop extensions and duals of the Brunn-Minkowski theory, which lies at the very core of convex geometric analysis. This includes ongoing investigations of the partial differential equations that arise in these extensions. Generalizations of affine functionals associated with convex bodies are another central focus of the proposed work. A large part of the project concerns establishing sharp affine isoperimetric isoperimetric inequalities and related analytic inequalities. The investigators will also continue their efforts to understand better the connections between affine convex geometric analysis and information theory. This project has potential for broader impact in many areas of science and engineering, because it deals with when and how can one reconstruct or approximate a geometric object from a limited number of geometric measurements. This is the central question in many practical endeavors ranging from computer vision to medical imaging. Some of the questions being studied by the investigators are sufficiently concrete that they can be explained to and investigated by graduate, undergraduate, and even high school students.