Convex Body Shape Recovery via Geometric Measures and Inequalities

该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）给出的部分数据来恢复几何物体的边界形状。这些几何物体，类似于我们周围的物体，可以拥有边缘和顶点（如正方形），但可以复杂得多（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）以及许多数学以外的其他领域（如经济学）。此外，这些问题与其他数学领域密切相关，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，它们在视觉上是可以理解的，并且由积极的本科生来解决。最近提出的对偶闵可夫斯基问题和Lp对偶闵可夫斯基问题是引起广泛关注的两个闵可夫斯基型问题。通过合作，首席研究员对假设数据和凸体原点对称时的解有了很好的理解。在那里，最终解在很大程度上依赖于具有非光滑边界的凸体。对非对称情况的理解是必要的，也是目标之一。挑战在于识别和解决适当的优化问题。另一个目标是了解涉及两个主体的优化问题如何在这种情况下提供帮助。当考虑带衰减的天线反射器设计问题时，也可以观察到同样的现象。Aleksandrov-Fenchel不等式是凸几何中最深奥的结果之一，包含了许多等周不等式。等式条件仍然是相当神秘的，因为即使在简单的情况下，具有分形边界结构的物体也可以作为极端情况出现。PI将研究混合区域测度的支持集的特征，这对于理解相等条件至关重要。该奖项反映了美国国家科学基金会的法定使命，并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，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.