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

这个研究项目关注的是从间接测量或一组设计要求中构造、识别或描述几何对象。这样的构造问题出现在数学、科学、工程和医学中。识别人体内的器官和肿瘤就是这样一个例子。雷达天线的形状设计是另一个问题。Brunn-Minkowski理论是核心学科之一的核心，它提供了用于此类应用的数学和计算工具。Brunn-Minkowski理论的核心是研究物体的几何测量类型（如表面积，体积，边界曲率）。另一个中心领域是闵可夫斯基问题，它询问一个物体是否可以从一组这些几何测量中重建。调查人员将继续努力扩大和丰富这两个中心领域。他们还将继续他们的工作，将信息论中的思想与Brunn-Minkowski理论联系起来。研究人员的工作产生了一些有趣的问题，有些问题仍然经受住了最好的研究数学家的努力，还有一些问题甚至可以由高中生和本科生探索。最近发现的双曲率测度导致了新的引人注目的问题被攻击。其中之一是对偶Minkowski问题，它需要求解一个新的完全非线性退化偏微分方程。通过结合几何和分析技术，研究人员（与各种合作者）一直在开发新的方法来解决与退化偏微分方程相关的几何测量的表征问题。这些新开发的技术（调查员）获得精细的估计积分的措施，导致发现（调查员），“测量浓度”是关键的现象显示的解决方案，这些几何特征的问题。继续这些调查应使调查人员在表征几何措施的基本问题上取得重大进展。仿射等周不等式多年来一直是Investigatoirs努力的中心焦点，并将探索一些新的方向。仿射等周不等式和尖锐的仿射Sobolev不等式之间的联系的研究（由研究者开创）显示出很大的希望，并将进一步探索。研究人员将继续利用他们的二次Brunn-Minkowski理论和信息理论9（来自电气工程）之间的联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

This research project concerns constructing, identifying, or describing geometric objects from either indirect measurements or from a set of design requirements. Such construction problems arise in mathematics, science, engineering, and medicine. Identifying organs and tumors within the human body is one such example. The design of the shapes of radar antennas is another. The Brunn-Minkowski theory is at the heart of one of the core subjects that provides the mathematical and computational tools used in such applications. The core of the Brunn-Minkowski theory concerns the study of types of geometric measurements (such as the surface area, volume, curvature of its boundary) of an object. Another central area are Minkowski problems, which ask whether an object can be reconstructed from a set of these geometric measurements. The investigators will continue their efforts to expand and enrich both of these central areas. They will also continue their work connecting ideas in information theory with Brunn-Minkowski theory. The work of the investigators has generated interesting questions, some that have still withstood the efforts of the best research mathematicians and other problems that can be explored even by high school and undergraduate students.The recent discovery of the dual curvature measures has led to new compelling problems to be attacked. One of them is the dual Minkowski problem which requires solving a novel fully nonlinear degenerated partial differential equations with measure data. By combining techniques from geometry and analysis, the Investigators (with various collaborators) have been developing new methods to solve characterization problems for geometric measures that are related to degenerated partial differential equations with measure data. These newly developed techniques (of the Investigators) of obtaining delicate estimates for integrals with respect to measures have led to the discovery (by the Investigators) that "measure concentration" is the key phenomenon displayed by solutions to those geometric characterization problems. Continuing these investigations should enable the Investigators to make significant progress on fundamental problems regarding characterizing geometric measures. Affine isoperimetric inequalities have for many years been a central focus of the Investigatoirs' efforts and a number of new directions are to be explored. The study (pioneered by the Investigators) of connections between affine isoperimetric inequalities and sharp affine Sobolev inequalities shows much promise and will be further explored. The Investigators will continue to exploit connections between their quadratic Brunn-Minkowski theory and the subject of information theory 9from electrical engineering).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.