几类特殊几何体的迷向常数  

Isotropic Constant of Some Special Geometric Bodies

作　　者：吴力荣[1] 何斌吾[1] 

机构地区：[1]上海大学理学院

出　　处：《上海大学学报（自然科学版）》2007年第1期41-46,共6页Journal of Shanghai University:Natural Science Edition

基　　金：上海市教委基金资助项目(214667);国家自然科学基金资助项目(10671119)

摘　　要：设K是Rn中体积为1,质心在原点的凸体,LK是它的迷向常数,寻找LK的上确界,是Banach空间局部理论(现代几何分析)中著名的未解决问题.目前最好的上界估计是LK<cn1/4logn,它是由Bourgain证明的.最近,何斌吾、冷岗松又证明了当r1Bn2 K r2Bn2(r1≥1/2,r2≤n/2)时,LK≤1/(2 3),并猜测在对称几何体中以超立方体的迷向常数为最大,在非对称几何体中以单形的迷向常数为最大.给出了在三维空间中全部正多面体的迷向常数的数值,从而说明这一猜测对三维空间中的正多面体是正确的.Let K∈Rn be a convex body with volume ｜ K ｜ = 1 and center of mass at the origin. Lk is an isotropic constant of K. Finding the least upper bound of Lk is a well-known open problem of Banach space local theory. Now the best estimate of upper bound is Lk〈 cn1/4logn, proved by Bourgain. Recently, He Bin-wu and Leng Gang-song have proved that Lk≤ 1/（2√3） when r1 B2→∩K→∩r2 B2n （ r1≥ 1/2, r2 ≤√n/2, and guess the isotropic constant of hypercube is the maximum among the symmetry geometric bodies, and one of simplex is the maximum among the asymmetrical geometric bodies. All isotropic constants of regular polyhedron in a three dimensional space are given in this paper. The result shows that the guess is right in a three dimensional space.

关 键 词：凸体 迷向体 迷向常数 Bourgain问题 

分 类 号：O186.5[理学—数学]