CAREER: Isoperimetric and Minkowski Problems in Convex Geometric Analysis

等周问题和Minkowski问题是凸几何分析的两个核心内容。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。这两类问题是内在联系在一起的。这个项目将利用这种联系，在各种情况下寻找等周问题或Minkowski问题的答案，当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时。尽管这些问题源于几何背景，但它们的应用超越了数学，进入了工程和设计领域，包括天线反射面设计和城市规划等领域。首席调查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，有高中教师和学生以及本科生和研究生参加。这些活动和研讨会旨在向学生展示首席调查者研究和一般数学研究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。原始对称情况下对偶Minkowski问题(表征对偶曲率度量)的解的存在在很大程度上已通过几何和分析的技术得到解决(由主要调查者和他的合作者解决)。这自然导致了与对偶Minkowski问题有关的等周问题的猜想。这种猜想的等周不等式也与许多其他凸性猜想背后的一个有趣的问题有关：某些对称如何改进估计？首席研究员还将研究来自仿射几何的Minkowski问题和等周不等式。这些等周不等式的特例与Almgren-Lieb的尖分式Sobolev不等式的仿射形式有关。研究这些问题所涉及的技术主要有凸几何分析和偏微分方程组。在过去的几十年里(特别是在过去的二十年里)，整个社会都在努力将凸体理论的结果推广到对数凹函数空间中的相应结果。在这个项目中，首席研究人员还将继续他过去的工作，将对偶曲率测量，他们的Minkowski问题，以及相关的等周不等式扩展到对数凹函数空间。这一奖项反映了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。

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.