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

这项研究项目涉及从间接测量或根据一组设计要求来构造、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例子。布鲁恩-明科夫斯基理论是核心学科之一的核心，它提供了在此类应用中使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(如物体的表面积、体积、边界的曲率)。另一个中心领域是Minkowski问题，它询问是否可以根据一组几何测量重建对象。调查人员将继续努力扩大和丰富这两个中心地区。他们还将继续将信息论中的思想与布鲁恩-明科夫斯基理论联系起来。研究人员的工作产生了一些有趣的问题，其中一些仍然经受住了最好的研究数学家的努力，以及其他即使是高中生和本科生也可以探索的问题。最近发现的双曲率测量导致了新的令人信服的问题需要攻击。其中之一是对偶Minkowski问题，它需要求解一个新的带测量数据的完全非线性退化偏微分方程组。通过结合几何和分析的技术，研究人员(与不同的合作者)一直在开发新的方法来解决几何测量的表征问题，这些几何测量与带有测量数据的退化偏微分方程组有关。这些新开发的技术(研究人员)获得关于测量的积分的精细估计导致(研究人员)发现“测量集中”是这些几何表征问题的解决方案所显示的关键现象。继续这些研究应该能够使研究人员在有关刻画几何测量的基本问题上取得重大进展。多年来，仿射等周不平等一直是调查工作的中心重点，并将探索若干新的方向。这项研究(由研究人员首创)关于仿射等周不等式和尖锐仿射Sobolev不等式之间的联系，显示出很大的前景，并将得到进一步的探索。调查人员将继续探索他们的二次Brunn-Minkowski理论与信息论主题9(来自电气工程)之间的联系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.