Isoperimetric Inequalities

等周不等式

• 批准号: 0104363
• 负责人: Erwin Lutwak
• 金额: $ 27.85万
• 依托单位: Polytechnic University of New York
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104363&HistoricalAwards=false
• 关键词: Isoperimetric; Inequalities

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.

DMS -0104363摘要研究人员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，并利用它们获得新的解析不等式。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，已经包含了许多最重要的方面，欧几里德angeometrics，包括尖锐的等周和Sobolev不等式。大部分的调查人员的努力将致力于发展和理解Brunn-Minkowski理论及其推广。调查人员的工作涉及一些最基本的对象在数学：机构和功能的普通欧几里德空间。这些被用来表示科学和工程中的真实的物体和现象。虽然无法预测这项工作的未来影响，但研究人员认为，该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Banach空间）到更多应用领域（如机器人视觉，信息论和体视学）的领域中是有用的。