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

该研究项目涉及通过间接测量或一组设计要求来构建几何形状。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，设计雷达天线或喷气式飞机的形状。布伦-闵可夫斯基理论是此类应用中使用的数学和计算工具的核心。布伦-闵可夫斯基理论的核心是等周不等式，它表达了一个物体的不同类型的几何测量（如表面积和体积）之间的关系，以及闵可夫斯基问题，它询问是否可以从一组几何测量（如边界的曲率）中重建形状。研究人员将继续他们的工作，将信息论中的思想与布伦-闵可夫斯基理论联系起来。这项工作产生了有趣的问题，高中生和本科生都可以探索这些问题。研究生也参与了这项研究。研究的总体主题是发展经典布伦-闵可夫斯基理论的延伸和对偶。在一个项目中，研究人员的目标是建立在先前的结果之上，表明对于实参数p的每个值，存在一个Lp布伦-闵可夫斯基理论。正如表面积测度的Minkowski问题是经典布伦-闵可夫斯基理论的中心焦点一样，Lp表面积测度的Minkowski问题也是Lp布伦-闵可夫斯基理论的中心问题。虽然大多数工作仅限于参数p大于1时，但研究人员和合作者正在探索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布伦-闵可夫斯基理论中费德勒曲率测量的类似物。令人惊讶的是，这个新发现的家族包括两个已知但重要的案例：锥体积测量和亚历山德罗夫的积分曲率测量。对于每一个这样的对偶曲率测度，都有一个对应的对偶Minkowski问题，该问题是一个完全非线性椭圆型偏微分方程。研究者的目的是进一步探讨对偶闵可夫斯基问题。第三个项目进一步研究了Aleksandrov积分曲率的Lp类似物；研究者的目标是彻底解决与这些措施相关的闵可夫斯基问题。研究人员将继续努力扩展先前关于信息论与Lp理论和双重布伦-闵可夫斯基理论之间的相似之处的工作。研究人员还将继续他们的工作，以建立对数-布伦-闵可夫斯基不等式，到目前为止，只在平面上建立了这个不等式。

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.