CAREER: Isoperimetric and Minkowski Problems in Convex Geometric Analysis

等环问题和闵可夫斯基问题是凸几何分析中的两个核心问题。前者比较几何测量（如体积和表面积），而后者使用这些测量的本地版本恢复几何图形的形状。这两类问题具有内在的联系。本项目将利用这种联系，在各种情况下寻找等周问题或闵可夫斯基问题的答案，其中一个问题的答案存在，而另一个问题的答案仍然难以捉摸。虽然这些问题起源于几何背景，但它们的应用范围从数学扩展到工程和设计，包括天线反射器设计和城市规划等领域。首席研究员将在当地科学博物馆、社区中心和学校组织一系列活动和研讨会，涉及高中教师和学生以及本科生和研究生。这些活动和研讨会旨在向处于教育生涯早期的学生展示首席研究员的研究和数学的有趣和探索的一面，提高社会对数学的认识和兴趣，并在历史上代表性不足的人群中推广数学。在原始对称情况下，对偶闵可夫斯基问题（表征对偶曲率测量）的解的存在性（由首席研究员和他的合作者）已经通过几何和分析技术在很大程度上解决了。这自然导致了与对偶闵可夫斯基问题相关的等周问题的猜想。这种推测的等周不等式也与许多其他凸性猜想背后的一个有趣问题有关：某些对称性如何改善估计？首席研究员还将研究来自仿射几何的闵可夫斯基问题和等周不等式。这些等周期不等式的特殊情况与Almgren-Lieb的尖锐分数Sobolev不等式的仿射版本有关。研究这些问题所涉及的技术来自凸几何分析和偏微分方程。在过去的几十年里（尤其是最近的二十年），学界一直在努力将凸体理论的结果扩展到它们在对数凹函数空间中的对应结果。在这个项目中，首席研究员还将继续他过去的工作，将对偶曲率测度，它们的闵可夫斯基问题，以及相关的等周不等式扩展到对数凹函数空间。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。

Isoperimetric problems and Minkowski problems are two central ingredients in Convex Geometric Analysis. The former compares geometric measurements (such as volume and surface area) while the latter recovers the shape of geometric figures using local versions of these measurements. The two types of problems are inherently connected. This project will exploit this connection to seek answers to either isoperimetric problems or Minkowski problems in various settings when answers to one exist while answers to the other remain elusive. Although these problems originate from a geometric background, their applications extend beyond mathematics into engineering and design, including areas like antenna reflector design and urban planning. The principal investigator will organize a series of events and workshops at local science museums, community centers, and schools, involving high school teachers and students as well as undergraduate and graduate students. These events and workshops aim to expose the fun and exploratory side of the principal investigator’s research and mathematics in general to students early in their educational careers, raise society's awareness and interest in mathematics, and promote mathematics among historically underrepresented populations.The existence of solutions to the dual Minkowski problem (that characterizes dual curvature measures) in the original symmetric case has been largely settled (by the principal investigator and his collaborators) through techniques from geometry and analysis. This naturally leads to conjectures involving isoperimetric problems connected to the dual Minkowski problem. Such conjectured isoperimetric inequalities are also connected to an intriguing question behind many other conjectures in convexity: how does certain symmetry improve estimates? The principal investigator will also study Minkowski problems and isoperimetric inequalities coming from affine geometry. Special cases of these isoperimetric inequalities are connected to an affine version of the sharp fractional Sobolev inequalities of Almgren-Lieb. The techniques involved in studying these questions are from Convex Geometric Analysis and PDE. In the last few decades (particularly the last two), there has been a community-wide effort to extend results in the theory of convex bodies to their counterparts in the space of log-concave functions. In this project, the principal investigator will also continue his past work to extend dual curvature measures, their Minkowski problems, and associated isoperimetric inequality to the space of log-concave functions.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.