Isoperimetric Inequalities

摘要奖：DMS 1312181，首席研究员：Erwin Lutwak，Deane Yang，Gao Yang Zhang这项提议旨在发展经典的Brunn-Minkowski理论的扩展和对偶。其中一个PI已经证明，对于每个大于1的p，都存在Lp Brunn-Minkowski理论。到目前为止，关于这一新理论的大部分工作都局限于p大于1，但PI和Boroczky在p=0的奇异情况下取得了进展。正如经典的Minkowski问题是经典Brunn-Minkowski理论的中心焦点一样，它的LP类比也是新理论的中心。对于p大于1的椭圆型偏微分方程解，现在称为Lp Minkowski问题，已经有了很大的进展。第零个Minkowski问题的彻底解决将对一些重要问题产生深远影响。PI与Boroczky合作，建立了当规定的“数据”为偶数时，这个对数Minkowski问题存在解的充要条件。它们还在两个维度上确立了独特性。在更高维度上证明唯一性是拟议工作的一部分。PI研究的一个特别焦点是仿射等周不等式的发展。多年来，PI已经建立了许多尖锐的仿射等周不等式及其解析对应物。私人投资促进机构最近开发的新方法将用于扩展先前的工作。虽然Brunn-Minkowski理论是解决涉及凸体到子空间的投影数据的各种基本逆问题的有效工具，但由PI之一发起的对偶Brunn-Minkowski理论的研究对于涉及凸体与子空间相交的对偶问题是理想的。PI们最近发现了在对偶理论中自然产生的新的对偶曲率度量，并相信对偶理论中新开发的工具可以用来解决PDE，这是Christoffel-Minkowski问题的自然对偶模拟。为解决这一偏微分方程所作的密集努力是拟议工作的核心部分。PI以前的工作已经表明了信息论和LP Brunn-Minkowski理论及其对偶之间有趣的相似之处。PI将继续调查与电气工程相关的学科与被认为是纯数学的学科之间的联系。Brunn-Minkowski理论及其对偶是凸几何分析的核心，也是几何层析的基础。几何层析成像的目的是从有关几何对象的低维截面或投影的数据中检索有关该几何对象的信息。它已经并显然将继续在科学、工程甚至医学(想想CAT扫描仪)中有实际应用。虽然等周不等式可以追溯到古希腊人，但许多较新的不等周不等式适用于几何、分析甚至工程学。私人投资机构建议建立(和扩展)的新的不平等和理论应该导致数学工具的发展，这些工具在数学、科学和工程中提供潜在的新应用。

AbstractAward: DMS 1312181, Principal Investigator: Erwin Lutwak, Deane Yang, Gaoyang ZhangThis proposal aims to develop both extensions and duals of the classical Brunn-Minkowski theory. One of the PIs has shown that for each p greater than 1 there is an Lp Brunn-Minkowski theory. Thus far, most work on this new theory has been limited to p greater than 1, but the PIs together with Boroczky have made progress in the singular case of p = 0. Just as the classical Minkowski problem is a central focus of the classical Brunn-Minkowski theory, its Lp analogue is central to the new theory. For p greater than 1, there has been significant progress on the elliptic PDE now known as the Lp Minkowski problem. A complete solution of the zero-th Minkowski problem would have a profound impact on a number of important questions. The PIs in collaboration with Boroczky have established both necessary and sufficient conditions for the existence of solutions to this logarithmic Minkowski problem when the prescribed "data"' is even. They have also established uniqueness in two dimensions. Proving uniqueness in higher dimensions is part of the proposed work. One particular focus of the PIs' research is the development of affine isoperimetric inequalities. Over the years the PIs have established a number of sharp affine isoperimetric inequalities and their analytic counterparts. New methods recently developed by the PIs will be used to extend earlier work. While the Brunn-Minkowski theory has been an effective tool for solving a variety of basic inverse problems involving data about projections of convex bodies onto subspaces, the study of a dual Brunn-Minkowski theory initiated by one of the PIs is ideal for dual questions involving intersections of convex bodies with subspaces. The PIs have recently discovered new dual curvature measures that arise naturally within the dual theory and believe that newly developed tools within the dual theory can be used to solve the PDE that is the natural dual analogue of the Christoffel-Minkowski problem. An intensive effort to solve this PDE is a central part of the work proposed. Previous work of the PIs has indicated fascinating parallels between information theory and both the Lp Brunn-Minkowski theory and its dual. The PIs will continue to investigate connections between a subject that is associated with Electrical Engineering and one that is considered pure mathematics.The Brunn-Minkowski theory and its dual are the core of convex geometric analysis and are the foundation of Geometric Tomography. Geometric Tomography aims at retrieving information about a geometric object from data about its lower dimensional sections or projections. It has had and clearly will continue to have practical applications in science, engineering, and even medicine (think CAT scan machines). While isoperimetric inequalities go back to the ancient Greeks, many of the newer ones are applicable in geometry, analysis, and even engineering. The new inequalities and theories that the PIs propose to establish (and extend) should result in the development of mathematical tools that offer potential new applications to mathematics, science, and engineering.