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Shape Discovery for Convex Bodies: Measures, Invariants, and Applications

凸体的形状发现：测量、不变量和应用

基本信息

批准号：
1710450
负责人：
Erwin Lutwak
金额：
$ 45.9万
依托单位：
New York University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710450&HistoricalAwards=false
关键词：
Shape Discovery Convex Bodies Measures

项目摘要

This research project concerns constructing a geometric shape from either indirect measurements or a set of design requirements. Such construction problems arise not only in mathematics, but also in science, engineering, and medicine. Examples include identifying organs and tumors in the human body or designing the shapes of radar antennas or jet airplanes. The Brunn-Minkowski theory is at the heart of the mathematical and computational tools used in such applications. Central to the Brunn-Minkowski theory are isoperimetric inequalities, which express relationships between different types of geometric measurements (such as the surface area and volume) of a body, and Minkowski problems, which ask whether a shape can be reconstructed from a set of geometric measurements (such as the curvature of the boundary). The investigators will continue their work connecting ideas in information theory with Brunn-Minkowski theory. This work has generated interesting questions that are accessible to and can be explored by high school and undergraduate students. Graduate students also are involved in the research.The overall theme of the research is to develop both extensions and duals of the classical Brunn-Minkowski theory. In one project, the investigators aim to build upon prior results showing that, for each value of a real parameter p, there is an Lp Brunn-Minkowski theory. Just as the Minkowski problem for the surface area measure is a central focus of the classical Brunn-Minkowski theory, the Minkowski problem for Lp surface area measure is central to the Lp Brunn-Minkowski theory. While most work been limited to when the parameter p is greater than 1, the investigators and collaborator are exploring the singular case, when p is 0, and aim to extend the work to non-positive p. A second project concerns analogues of Federer's curvature measures within the dual Brunn-Minkowski theory. Surprisingly, this newly discovered family includes two already known but important cases: the cone-volume measure and Aleksandrov's integral curvature measure. For each such dual curvature measure, there is a corresponding dual Minkowski problem, which is a fully nonlinear elliptic partial differential equation. The investigators aim to further explore the dual Minkowski problem. A third project further investigates Lp analogues of Aleksandrov's integral curvature; the investigators aim to solve completely the Minkowski problems associated with these measures. The investigators will continue their efforts to extend previous work on the parallels between information theory and both the Lp and dual Brunn-Minkowski theories. The investigators will also continue their work toward establishing the log-Brunn-Minkowski inequality, which to date has been established only in the plane.

这项研究项目涉及从间接测量或一组设计要求构建几何形状。这样的构造问题不仅出现在数学中，也出现在科学、工程和医学中。例如，识别人体器官和肿瘤，或设计雷达天线或喷气式飞机的形状。布鲁恩-明科夫斯基理论是此类应用中使用的数学和计算工具的核心。Brunn-Minkowski理论的核心是等周不等式，它表达了物体不同类型的几何测量(如表面积和体积)之间的关系，以及Minkowski问题，它询问是否可以根据一组几何测量(如边界的曲率)重建形状。调查人员将继续他们的工作，将信息论中的思想与布鲁恩-明科夫斯基理论联系起来。这项工作产生了一些有趣的问题，高中生和本科生可以接触到，也可以探索。研究生也参与了这项研究。研究的总体主题是发展经典的Brunn-Minkowski理论的延伸和对偶。在一个项目中，研究人员的目标是建立在先前结果的基础上，该结果表明，对于真实参数p的每个值，都存在Lp Brunn-Minkowski理论。正如表面积度量的Minkowski问题是经典Brunn-Minkowski理论的中心焦点一样，LP表面积度量的Minkowski问题也是LP Brunn-Minkowski理论的中心问题。虽然大多数工作仅限于当参数p大于1时，但研究人员和合作者正在探索当p为0时的奇异情况，并旨在将工作扩展到非正p。第二个项目涉及对偶Brunn-Minkowski理论中的费德勒曲率度量的类似。令人惊讶的是，这个新发现的族包括两个已知但重要的情况：锥体体积测量和亚历山大罗夫的积分曲率测量。对于每一个这样的对偶曲率度量，都有一个对应的对偶Minkowski问题，它是一个完全非线性的椭圆型偏微分方程。调查人员的目标是进一步探索双重Minkowski问题。第三个项目进一步研究Aleksandrov积分曲率的Lp类似；调查者的目标是完全解决与这些度量相关的Minkowski问题。调查人员将继续努力扩展之前关于信息论与有限合伙人理论和对偶Brunn-Minkowski理论之间的相似之处的工作。调查人员还将继续他们的工作，以建立LOG-Brunn-Minkowski不等式，到目前为止，该不等式仅在平面上建立。