Isoperimetric Inequalities

摘要：DMS - 0104363的研究重点是建立在线性或仿射变换下不变的尖锐几何不等式，并利用它们得到新的解析不等式。由于仿射结构不包含距离或角度的概念，人们可能会认为没有什么可说或可做的。事实上，有一个丰富而深刻的理论已经涵盖了欧几里得几何的许多最重要的方面，包括尖锐的等周不等式和索博列夫不等式。大多数研究者的努力将致力于发展和理解布伦-闵可夫斯基理论及其推广。研究人员的工作涉及数学中一些最基本的对象：普通欧几里得空间中的物体和函数。它们在科学和工程中用来表示真实的物体和现象。虽然不可能预测这项工作的未来影响，但研究人员相信，在这个项目中开发的概念和技术可以证明在从纯数学领域（如微分几何、偏微分方程和巴拿赫空间）到更多应用领域（如机器人视觉、信息论和立体学）的领域是有用的。

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.