Applications of Analytic and Probabilistic Methods in Convexity to Geometric Functionals

解析和概率方法在几何泛函凸性中的应用

• 批准号: RGPIN-2022-02961
• 负责人: Tatarko, Kateryna
• 金额: $ 1.38万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=764420
• 关键词: Applications; Analytic; Probabilistic; Methods; Convexity

The proposed research program belongs to the related fields of convex geometry, geometric functional analysis and asymptotic geometric analysis. These are very active research areas that interact with each other and that are closely connected to many other fields such as differential geometry, harmonic analysis, probability theory and combinatorics. Developments in these areas have found and continue to find applications in mathematical physics, information theory, data science, geophysics, computer science among others. One of the central elements of convexity theory is the so-called Steiner formula which states that the volume of convex bodies behaves as a polynomial. The coefficients that arise in this polynomial are classical geometric invariants, called intrinsic volumes or quermassintegrals, the study of which goes back to Hermann Minkowski. The main goal of this proposal is the study of functionals associated with convex bodies, including quermassintegrals and their generalizations, in various settings. I plan to use techniques from geometric functional analysis in combination with probabilistic tools and others to approach the problems. The proposal consists of three main parts. The first topic is related to the study of the Lp Steiner formula via affine surface area. The second part is about isoperimetric and reverse isoperimetric problems. The last part concerns stochastic counterparts of inequalities for functionals associated with convex bodies.

所提出的研究项目属于凸几何、几何泛函分析和渐近几何分析的相关领域，这些领域是非常活跃的研究领域，彼此相互作用，并且与微分几何、调和分析、概率等许多其他领域密切相关。这些领域的发展已经并继续在数学物理学、信息论、数据科学、地球物理学、计算机科学等领域找到应用，凸性理论的核心要素之一是所谓的斯坦纳公式。凸体的体积表现为多项式。该多项式中出现的系数是经典的几何不变量，称为固有体积或 quermassintegrals，其研究可以追溯到赫尔曼·明科夫斯基（Hermann Minkowski）。该提案的主要目标是研究相关的泛函。我计划在各种设置中使用几何泛函分析技术与概率工具和其他工具相结合来处理凸体，包括 quemassintegrals 及其概括。该提案由三个主要部分组成。第一个主题与通过仿射表面积研究 Lp Steiner 公式有关，最后一部分涉及相关函数的不等式的随机单位。具有凸体。