Isotropic bodies and Bourgain's problem

作     者：HE Binwu & LENG Gangsong Department of Mathematics, Shanghai University, Shanghai 200444, China 

作者机构：1. Department of Mathematics Shanghai University 200444 Shanghai China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2005年第48卷第5期

页      面：666-679页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：This work was supported by the National Natural Science Foundation of China(Grant No.10271071) 

主　　题：convex body isotropic body isotropic constant Bourgain’s problem spherical section function. 

摘      要：Let K (?) Rn be a convex body of volume 1 whose barycenter is at the origin, LK be the isotropic constant of K. Finding the least upper bound of LK , being called Bourgain s problem, is a well known open problem in the local theory of Banach space. The best estimate known today is LK cn1/4 log n, recently shown by Bourgain, for an arbitrary convex body in any finite dimension. Utilizing the method of spherical section function, it is proven that if K is a convex body with volume 1 and r1Bn2 (?) K (?) r2Bn2,(r1≥1/2, r2≤(?)/2), then (?) ≤ (?) and find the conditions with equality. Further, the geometric characteristic of isotropic bodies is shown.