Convex Body Shape Recovery via Geometric Measures and Inequalities

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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.