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 ff and only ff this simplex is regular and that the John ellipsoid of a regular simplex is its inscribed ball.
The main purpose of the present article is to establish some new strengthed and reversed Pachpatte's type inequalities. As applications, some new type Hilbert's inequlities are generalized and strengthened.
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.
HE Binwu & LENG Gangsong Department of Mathematics, Shanghai University, Shanghai 200444, China
