CAREER: Isoperimetric and Minkowski Problems in Convex Geometric Analysis

等周问题和Minkowski问题是凸几何分析中的两个重要组成部分。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。这两类问题有着内在的联系。这个项目将利用这种联系，以寻求答案，无论是等周问题或闵可夫斯基问题在各种设置时，一个答案存在，而另一个答案仍然难以捉摸。虽然这些问题起源于几何背景，但它们的应用超出了数学，进入了工程和设计，包括天线反射器设计和城市规划等领域。主要研究者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，参与者包括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学生在其教育生涯的早期展示首席研究员的研究和数学的乐趣和探索性，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。对偶Minkowski问题解的存在性（表征双曲率措施）在原来的对称情况下，已在很大程度上解决（由主要研究者和他的合作者）通过技术从几何和分析。这自然会导致涉及到对偶闵可夫斯基问题的等周问题。这种约束的等周不等式也与许多其他凸性约束背后的一个有趣的问题有关：某些对称性如何改善估计？主要研究者还将研究闵可夫斯基问题和来自仿射几何的等周不等式。这些等周不等式的特殊情况下连接到一个仿射版本的尖锐分数Sobolev不等式的Almante-Lieb。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（特别是过去的二十年），有一个社区范围内的努力，以扩大在凸体理论的结果，他们的对手在对数凹函数的空间。在这个项目中，首席研究员还将继续他过去的工作，以扩大双曲率措施，他们的闵可夫斯基问题，并相关的等周不等式的对数凹函数的空间。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.