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。Brunn-Minkowski理论是这些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，它表达了物体的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Brunn-Minkowski理论联系起来。这项工作产生了有趣的问题，可以访问，并可以由高中和本科生探索。研究生也参与了这项研究。研究的总主题是发展经典Brunn-Minkowski理论的扩展和改进。在一个项目中，研究人员的目标是建立在先前的结果表明，对于每个值的真实的参数p，有一个Lp Brunn-Minkowski理论。正如表面积测度的闵可夫斯基问题是经典Brunn-Minkowski理论的中心焦点一样，Lp表面积测度的闵可夫斯基问题也是Lp Brunn-Minkowski理论的中心。虽然大多数工作被限制在当参数p大于1时，研究人员和合作者正在探索奇异的情况下，当p为0时，并旨在将工作扩展到非正p。令人惊讶的是，这个新发现的家庭包括两个已知的，但重要的情况下：锥体积措施和亚历山德罗夫的积分曲率措施。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，这是一个完全非线性椭圆型偏微分方程。研究人员的目标是进一步探索对偶闵可夫斯基问题。 第三个项目进一步调查的LP类似物亚历山德罗夫的积分曲率;调查人员的目的是完全解决闵可夫斯基问题与这些措施。研究人员将继续努力，以扩大以前的工作之间的相似性信息理论和LP和双布伦-闵可夫斯基理论。研究人员还将继续建立log-Brunn-Minkowski不等式的工作，迄今为止，该不等式仅在平面上建立。

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.