Probabilistic Approach in the Local Theory of Banach Spaces and Convex Geometry

9706835鲁德尔森概率方法构成了凸几何和Banach空间局部理论的一个非常迅速发展的领域。这一领域的最新成果尤其与TALAGRAND提出的优化措施方法有关。这种方法能够解决组合学、调和分析和凸几何中的几个深层次问题。这项工作将继续研究优化措施的构造，并研究这种方法在局部理论和调和分析的具体问题中的可能应用。另一个方向与研究凸体有关，凸体不一定是对称的。这是一个新项目，其灵感来自于在将局部理论的结果推广到星形天体方面的最新进展。证明了p-凸体的局部理论最重要的结果，包括逆Santalo和Brunn-Minkowski不等式。与p-凸对称体不同，在一般凸体的情况下，几乎什么都不是已知的。适用于凸体和p-凸对称体的基本技术工具，如果考虑一般的凸体，则不再起作用。此外，关于非对称凸体的已知事实表明，它们的性质可能与对称体的性质有很大的不同，该项目的目的是建立一个与现有的凸对称体理论平行的一般凸体理论。凸几何是研究高维凸体性质的数学领域。这包括寻找具有某些良好属性的给定体的截面、计算体的体积、用具有更好结构的另一个凸体来近似凸体等。虽然这些问题一般不能通过显式构造来解决，但通常可以证明随机选择的截面或随机修改的体具有所需的属性。这种方法可以将解析和几何问题简化为对某一随机过程行为的研究。该项目将集中于开发一种分析这种过程的方法，并研究一般凸体的结构。所得到的结果适用于调和分析和Banach空间理论以及计算机科学中的各种问题，如构造有效的优化算法和快速计算凸体的体积。

9706835 Rudelson Probabilistic methods constitute a very rapidly developing field of Convex Geometry and Local Theory of Banach spaces. Recent results in this area are connected in particular to the majorizing measure approach, introduced by Talagrand. This method enabled the solution of several deep problems in Combinatorics, Harmonic Analysis, and Convex Geometry. This work will continue investigation of constructions of majorizing measures as well as study possible applications of this method to concrete problems of Local Theory and Harmonic Analysis. Another direction is related to the study of convex bodies which are not necessary symmetric. This is a new project inspired by recent progress in generalization of the results of the Local Theory to star-shaped bodies. It turned out to be possible to prove the most important results of the Local Theory, including inverse Santalo and Brunn-Minkowski inequalities, for p-convex bodies. Unlike p-convex symmetric bodies, in the case of general convex bodies almost nothing is known. The basic technical tools, which work for convex and p-convex symmetric bodies, do not work any more if general convex bodies are considered. Moreover, the known facts about non-symmetric convex bodies suggest that there might be a significant difference between their properties and the properties of symmetric bodies.The aim of the project is to construct a theory of general convex bodies which is parallel to the existing theory of convex symmetric bodies. Convex Geometry is a field of Mathematics which studies the properties of convex bodies of high dimension. This includes finding sections of a given body with certain nice properties, computing the volume of a body, approximating a convex body by another one having a better structure, etc. While these problems cannot be solved in general by an explicit construction, it is often possible to show that a randomly chosen section or a random modification of a body has the desired prop erties. This approach enables the reduction of the analytic and geometric problems to the investigation of the behavior of a certain random process. The project will concentrate on developing a method to analyze such processes and on the investigation of the structure of general convex bodies. The results obtained in this direction are applicable to various problems in Harmonic Analysis and Banach Space theory as well as in Computer Science, like construction of effective optimization algorithms and fast computation of the volumes of convex bodies.