Applications of Analytic and Probabilistic Methods in Convexity to Geometric Functionals

所提出的研究方案属于凸几何、几何泛函分析和渐近几何分析的相关领域。这些都是非常活跃的研究领域，相互影响，并与许多其他领域密切相关，如微分几何、谐波分析、概率论和组合学。这些领域的发展已经并将继续在数学物理、信息理论、数据科学、地球物理、计算机科学等领域得到应用。凸性理论的核心要素之一是所谓的斯坦纳公式，该公式指出凸体的体积表现为多项式。在这个多项式中出现的系数是经典的几何不变量，称为内在体积或quermass积分，其研究可以追溯到赫尔曼·闵可夫斯基。本提案的主要目标是研究与凸体相关的泛函，包括在各种情况下的quermass积分及其推广。我计划使用几何泛函分析的技术，结合概率工具和其他方法来解决问题。该提案由三个主要部分组成。第一个课题是通过仿射表面积来研究Lp Steiner公式。第二部分是关于等周和逆等周问题。最后一部分涉及与凸体相关的泛函不等式的随机对应物。

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.