Isoperimetric Inequalities

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