Isoperimetric Inequalities

项目负责人：Erwin Lutwak， Deane Yang，高勇张本项目是一个长期项目的延续，研究凸几何分析的核心——Brunn-Minkowski理论的推广和对偶。这包括正在进行的对这些扩展中产生的偏微分方程的研究。与凸体相关的仿射泛函的泛化是所提出工作的另一个中心焦点。该项目的很大一部分涉及建立尖锐仿射等周不等式和相关的解析不等式。研究人员还将继续努力更好地理解仿射凸几何分析和信息论之间的联系。该项目在科学和工程的许多领域具有更广泛的影响，因为它涉及何时以及如何从有限数量的几何测量中重建或近似几何物体。从计算机视觉到医学成像，这是许多实际工作的核心问题。研究者正在研究的一些问题是足够具体的，他们可以解释和调查研究生，本科生，甚至高中生。

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.