Convex Body Shape Recovery via Geometric Measures and Inequalities

这个项目的中心主题是使用局部或全局几何测量(如面积和体积)的部分数据恢复几何对象的边界形状。这些几何对象，类似于我们周围看到的对象，可以拥有边和顶点(如正方形)，但也可以复杂得多(例如，拥有分形结构)。这种性质的问题出现在许多工程/设计问题(如天线反射器的设计)中，以及许多其他数学领域(如经济学)。此外，这些问题与数学的其他领域有很好的联系，包括偏微分方程和泛函分析。这些问题的另一个好处是，在各种特殊情况下，它们是有动机的本科生可以直观理解和解决的。最近提出的对偶Minkowski问题和Lp对偶Minkowski问题是两个备受关注的Minkowski类型问题。通过合作，首席研究人员已经很好地理解了当一个人假设数据和凸体的原点对称时的解。在那里，最终的解在很大程度上取决于具有非光滑边界的凸体。对非对称情况的理解是当务之急，也是目标之一。挑战在于找出并解决适当的优化问题。另一个目标是了解涉及两个实体的优化问题如何在此设置中有所帮助。当考虑考虑衰减的天线反射面设计问题时，也可以观察到同样的现象。亚历克桑德罗夫-芬切尔不等式是凸几何中最深刻的结果之一，它包含了许多等周不等式。相等性条件仍然非常神秘，因为即使在简单的情况下，具有分形边界结构的物体也可能作为极端情况出现。PI将研究混合区域措施支持集的特征，这对于理解平等条件至关重要。该奖项反映了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。

The central theme of this project is the recovery of boundary shape of a geometric object using partial data given as local or global geometric measurements (such as areas and volumes). These geometric objects, similar to the objects we see around us, can possess edges and vertices (such as a square), but can be so much more complicated (for example, possessing a fractal structure). Problems of this nature arise in many engineering/designing problems (such as the designing of antenna reflector) in addition to many other areas out of mathematics (such as economics). Moreover, these problems are well connected with other areas of mathematics, including PDE and functional analysis. Another benefit of these problems is that in various special cases, they are visually understandable and solvable by motivated undergraduate students.The recently posed dual Minkowski problem and Lp dual Minkowski problem are two Minkowski-type problems that received much attention. The principal investigator, through collaboration, has gained a good understanding of the solutions when one assumes origin-symmetry of the data and the convex body. There, the final solutions depend very much on convex bodies with non-smooth boundaries. An understanding in the non-symmetric case is imperative and is one of the goals. The challenge is to identify and solve the proper optimization problem. Another goal is to understand how an optimization problem involving two bodies can help in this setting. The same phenomenon can be observed when one considers the antenna reflector design problem with decay. The Aleksandrov-Fenchel inequality is one of the deepest results in convex geometry and emcompasses many isoperimetric inequalities. The equality condition remains quite mysterious as even in the simple cases, bodies with fractal boundary structures can arise as extremal cases. The PI will study the characterization of the support set of mixed area measures which is essential in understanding the equality condition.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.