Isoperimetric Inequalities

Brunn-Minkowski理论，通常被称为混合体积理论，是凸几何分析的核心。多年来，数学、科学和工程的其他领域也发现了许多应用，包括偏微分方程、代数几何、统计学、数论、闵可夫斯基和芬斯勒几何、体视学、信息论。这个建议的目标是扩展和扩展的经典Brunn-Minkowski理论的发展。一个特别的焦点是仿射等周不等式的发展。仿射不等式在偏微分方程、Banach空间几何、几何层析成像、甚至机器人视觉中有着广泛的应用。另一个特别的重点将是仿射等周不等式的分析对应的发展。由PI开发的新方法将用于尝试在一些长期困扰的仿射等周不等式上取得进展。该提案的另一个主要焦点涉及一个规定的曲率问题，称为Lp Minkowski问题，它在扩展的Brunn-Minkowski理论中自然产生。PI的工作表明，信息论和凸几何分析之间存在有趣的潜在联系。这些连接将被利用的目的是建立新的几何结果的启发，他们的信息theories people.The classical Brunn-Minkowski theory has provided powerful tools needed to solve a variety of basic inverse problems where the only“data”available involved information about the projections of convex bodies to low-dimensional subspaces（such as lines and planes）.经典的Brunn-Minkowski理论提供了强大的工具，需要解决各种基本的反问题，唯一的“数据”涉及的凸体到低维子空间（如线和平面）的投影信息。然而，经典Brunn-Minkowski理论的工具在处理这些问题的对偶时几乎没有价值，其中“子空间上的投影”被“与子空间的相交”所取代。 PI正在继续努力开发一个双Brunn-Minkowski理论。这个对偶理论已经导致了一个长期存在的问题的解决方案，其中已知的信息涉及未知物体与平面的相交。 这样的“逆问题”不仅是数学的基础，也是科学和工程的基础。

The Brunn-Minkowski theory, often called the theory of mixed volumes, is the very core of convex geometric analysis. Over the years numerous applications to---or connections with---other fields of mathematics, science, and engineering have been discovered, including partial differential equations, algebraic geometry, statistics, number theory, Minkowski and Finsler geometry, stereology, information theory. The goal of this proposal is the development of extensions and duals of the classical Brunn-Minkowski theory. A special focus is the development of affine isoperimetric inequalities. There have been numerous applications of affine inequalities in PDEs, Banach space geometry, geometric tomography, and even robot vision. Another special focus will be the development of the analytic counterparts of affine isoperimetric inequalities. New methods developed by the PIs will be used to attempt to make progress on a number of long-conjectured affine isoperimetric inequalities. Another main focus of the proposal concerns a prescribed curvature problem, known as the Lp Minkowski problem, that arises naturally within the extended Brunn-Minkowski theory. Work of the PIs indicates that there are interesting potential connections between information theory and convex geometric analysis. These connections will be exploited with the aim of establishing new geometric results inspired by their information theory counterparts.The classical Brunn-Minkowski theory has provided powerful tools needed to solve a variety of basic inverse problems where the only "data" available involves information regarding the projections of convex bodies onto lower-dimensional subspaces (such as lines and planes). However, the tools of the classical Brunn-Minkowski theory have been of little value in dealing with the dual of these questions, where "projections onto subspaces" are replaced by "intersections with subspaces". The PIs are continuing efforts to develop a dual Brunn-Minkowski theory. This dual theory has already led to the solution of a longstanding problem where the known information involved the intersections of the unknown bodies with planes. Such "inverse problems" are basic in not only mathematics but also science and engineering.