Isoperimetric Inequalities

项目负责人：Erwin Lutwak， Christoph Haberl,Deane Yang, Gaoyang zhang。本项目旨在发展经典Brunn-Minkowski理论（通常称为混合体积理论）在凸几何分析中的现代扩展。多年来，pi已经建立了一些基本的仿射等周不等式及其分析对应。新开发的方法（由pi和其他人）将被用来攻击长期猜测的不平等。其中一个pi展示了一个关于凸体的p和的旧概念如何导致p-布伦-闵可夫斯基理论的雏形。新理论中出现的p-不等式几乎总是比经典理论中出现的p-不等式更强。最近的进展表明，有必要在布伦-闵可夫斯基理论的发展中进入下一个进化步骤：奥尔利茨布伦-闵可夫斯基理论。最近，pi已经成功地从布伦-闵可夫斯基理论中发现了两个基本算子的Orlicz模拟，并为它们建立了基本的仿射等周不等式。Orlicz Brunn-Minkowski理论的发展将是PIs努力的主要焦点。正如经典闵可夫斯基问题是经典布伦-闵可夫斯基理论的中心焦点一样，它的p-类比也是p-布伦-闵可夫斯基理论的中心。椭圆型偏微分方程的Orlicz版本是Orlicz - Brunn-Minkowski理论的中心焦点，并将对该偏微分方程进行持续的攻击。研究凸体的投影和相交是一门基础学科，具有重要的实用价值。布伦-闵可夫斯基理论与谐波分析的余弦变换是研究投影的理想工具。凸几何中有一种迷人的对偶性。其中一位pi发起了对偶布伦-闵可夫斯基理论的研究，该理论与Radon变换一起是研究交叉点和解决长期开放问题的必要工具，正如其中一位pi和其他人先前的工作所表明的那样。提出了对偶布伦-闵可夫斯基理论的继续发展。pi和其他人的工作表明，信息论（通常与电气工程相关的学科）和凸几何分析之间存在有趣的联系。pi将继续探索这些学科之间的相互作用。布伦-闵可夫斯基理论是凸几何分析的核心，是几何层析成像和立体学等学科的基础。这些学科研究具有很强实践背景的数学问题，如测量肺的扩散能力，确定岩石中矿物质的体积，以及检测人体内的肿瘤。布伦-闵可夫斯基理论在数学、科学和工程领域得到了广泛应用。例如，从有关剖面的信息（想想CAT扫描机）重建隐藏对象涉及层析分析。所提出的工作将导致凸几何分析的新理论和新技术的发展，这是具有应用于科学和工程的潜在的新的数学工具。

AbstractAward: DMS-1007347Principal Investigator: Erwin Lutwak, Christoph Haberl,Deane Yang, Gaoyang ZhangThe aim of this project is to develop modern extensions of the classical Brunn-Minkowski theory (often called the theory of mixed volumes) in convex geometric analysis. Over the years, the PIs have established a number of fundamental affine isoperimetric inequalities and their analytic counterparts. Newly developed methods (by the PIs and others) will be exploited to attack long-conjectured inequalities. One of the PIs showed how an old notion of p-sum of convex bodies leads to an embryonic p-Brunn-Minkowski theory. The p-inequalities which arise in the new theory turn out to be almost invariably stronger than their classical counterparts. Recent advances have demonstrated the need to move to the next evolutionary step in the development of the Brunn-Minkowski theory: An Orlicz Brunn-Minkowski theory. Very recently, the PIs have succeeded in discovering the Orlicz analogue of two fundamental operators from the Brunn-Minkowski theory and have established the fundamental affine isoperimetric inequalities for them. The development of an Orlicz Brunn-Minkowski theory will be a main focus of the PIs efforts. Just as the classical Minkowski problem is a central focus of the classical Brunn-Minkowski theory, its p-analogue is central to the p-Brunn-Minkowski theory. The Orlicz versions of this elliptic partial differential equation are a central focus of the Orlicz Brunn-Minkowski theory and a sustained attack on this partial differential equation will be undertaken. The study of projections and intersections of convex bodies is fundamental and of significant practical value. The Brunn-Minkowski theory together with the cosine transform from harmonic analysis are ideal tools for the study of projections. There is a fascinating duality in convex geometry. One of the PIs initiated the study of the dual Brunn-Minkowski theory, which together with Radon transforms are the necessary tools for the study of intersections and to solve long-standing open problems as shown by previous work of one of the PIs and others. Continued development of the dual Brunn-Minkowski theory is proposed. Work of the PIs and others, indicates that there are interesting connections between Information Theory (a subject usually associated with Electrical Engineering) and convex geometric analysis. The PIs will continue their exploration of the interactions between these subjects.The Brunn-Minkowski theory is the core of convex geometric analysis and is the foundation of subjects such as geometric tomography and stereology. These subjects investigate mathematical problems with strong practical backgrounds such as measuring the diffusion capacity of the lung, determining the volume of minerals in rocks, and detecting tumors in the human body. Numerous applications of the Brunn-Minkowski theory have been discovered in mathematics, science and engineering. For example, the reconstruction of hidden objects from information regarding sections (think CAT scan machines) involves a tomographic analysis. The proposed work should result in the development of new theories and techniques in convex geometric analysis which are potential new mathematical tools with applications to science and engineering.