Isoperimetric Inequalities

Polytechnic University Of New York, Brooklyn NY

Investigators

Abstract

Abstract for DMS - 0104363 The investigators' research focuses on establishing sharp geometric inequalities that are invariant under linear or affine transformations and using them to obtain new analytic inequalities. Since an affine structure contains no notion of distance or angle, one might think that there is little to say or do. In fact, there is a rich and deep theory that already encompasses many of the most important aspects of Euclidean geometry, including the sharp isoperimetric and Sobolev inequalities. Most of the investigators' efforts will be devoted to developing and understanding the Brunn-Minkowski theory and its generalizations. The investigators' work involves some of the most fundamental objects in mathematics: bodies in and functions of ordinary Euclidean space. These are used to represent real objects and phenomena in science and engineering. Although it is impossible to predict the future impact of this work, the investigators believe that the concepts and techniques developed in this project could prove to be useful in fields ranging from pure mathematical areas such as differential geometry, partial differential equations, and Banach spaces to more applied areas such as robot vision, information theory, and stereology.