Isotropic bodies and Bourgain's problem

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.

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 ( ) r2 Bn2, (r1 ≥1/2, r2 ≤ -√n/2), then1/√2πe ≤ LK ≤ 1/2√3,and find the conditions with equality. Further,the geometric characteristic of isotropic bodies is shown.