In this paper, a description of the John contact points of a regular simplex was given. It was prove that the John ellipsoid of any simplex is ball if and only if this simplex is regular and that the John ellipsoid of a regular simplex is its inscribed ball.
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.
HE Binwu & LENG Gangsong Department of Mathematics, Shanghai University, Shanghai 200444, China
