Some Open Problems in Asymptotic Geometric Analysis
Some Open Problems in Asymptotic Geometric Analysis
复制标题
渐近几何分析中的一些开放问题
DOI：
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发表时间：
2018
影响因子：
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通讯作者：
E. Werner
中科院分区：
文献类型：
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作者：
B. Klartag;E. Werner
我们描述了四个相关的开放问题在渐近几何分析：超平面猜想，迷向常数猜想，西尔维斯特的问题，和单纯形猜想。高维系统在数学和应用科学中经常出现，理解高维现象变得越来越重要。渐近几何分析作为一个新的领域出现，处理正是这样的现象。它处于泛函分析、凸几何和概率论等学科的十字路口，也与数学物理和理论计算机科学有关。在过去的二十年里，这一领域取得了巨大的增长。超平面理论的一个主要推动力是超平面猜想或切片问题。受谐波分析中出现的问题的启发，它首先由J. Bourgain提出，并通过许多人的工作而闻名，例如K。Ball和V. Milman和A.帕约尔它询问是否体积为1的每个中心凸体都有一个通过原点的超平面截面，其体积大于绝对常数c > 0：超平面猜想。每个定心的体积凸体K| K| = 1具有通过原点的超平面截面，其体积大于绝对常数c > 0，与维数无关。Bo'az Klartag是魏茨曼研究所和特拉维夫大学的数学教授。他的电子邮件地址是klartagb@post.tau.ac.il。伊丽莎白·沃纳是凯斯西储大学的数学教授。她的电子邮件地址是elisabeth.werner@case
We describe four related open problems in asymptotic geometric analysis: the hyperplane conjecture, the isotropic constant conjecture, Sylvester’s problem, and the simplex conjecture. High-dimensional systems are frequent in mathematics and applied sciences, and understanding high-dimensional phenomena has become increasingly important. Asymptotic geometric analysis emerged as a new area that deals with exactly such phenomena. It is at the crossroads of such disciplines as functional analysis, convex geometry, and probability theory and bears connection to mathematical physics and theoretical computer science as well. The last two decades have seen tremendous growth in this area. A major impulse for the theory is the hyperplane conjecture or slicing problem. Motivated by questions arising in harmonic analysis, it was first formulated by J. Bourgain and made known through the work of many people, such as K. Ball and V. Milman and A. Pajor. It asks if every centered convex body of volume 1 has a hyperplane section through the origin whose volume is greater than an absolute constant c > 0: Hyperplane Conjecture. Every centered convex body K of volume |K| = 1 has a hyperplane section through the origin with volume greater than an absolute constant c > 0, independent of dimension. Bo’az Klartag is professor of mathematics at the Weizmann Institute and Tel Aviv University. His email address is klartagb@post .tau.ac.il. Elisabeth Werner is professor of mathematics at Case Western Reserve University. Her email address is elisabeth.werner@case