Isoperimetric Inequalities

奖：DMS-0405707主要研究员：Erwin Lutwak，Deane Yang，Gayong Zhang这个项目继续一个长期计划，以发展Brunn-Minkowski理论的扩展和对偶，该理论位于凸几何分析的非常核心。这包括对在这些推广中出现的偏微分方程的持续研究。与凸体相关的仿射函数的推广是所提出的工作的另一个中心焦点。该项目的很大一部分内容涉及建立尖锐的仿射等参等周不等式和相关的解析不等式。研究人员还将继续他们的研究，以更好地理解仿射几何分析和信息论之间的联系。这个项目可能会在许多科学和工程领域产生更广泛的影响，因为它涉及人们何时以及如何从有限数量的几何测量中重建或近似几何对象。这是从计算机视觉到医学成像的许多实际工作的中心问题。研究人员正在研究的一些问题非常具体，研究生、本科生甚至高中生都可以向他们解释和调查。

AbstractAward: DMS-0405707Principal Investigator: Erwin Lutwak, Deane Yang, Gaoyong ZhangThis project continues a long term program to develop extensionsand duals of the Brunn-Minkowski theory, which lies at the verycore of convex geometric analysis. This includes ongoinginvestigations of the partial differential equations that arisein these extensions. Generalizations of affine functionalsassociated with convex bodies are another central focus of theproposed work. A large part of the project concerns establishingsharp affine isoperimetric isoperimetric inequalities and relatedanalytic inequalities. The investigators will also continue theirefforts to understand better the connections between affineconvex geometric analysis and information theory.This project has potential for broader impact in many areas ofscience and engineering, because it deals with when and how canone reconstruct or approximate a geometric object from a limitednumber of geometric measurements. This is the central questionin many practical endeavors ranging from computer vision tomedical imaging. Some of the questions being studied by theinvestigators are sufficiently concrete that they can beexplained to and investigated by graduate, undergraduate, andeven high school students.