A concentration of mass property for isotropic convex bodies in high dimensions

It is shown that, for a certain subclass of istropic convex sets in R-n, th e mass concentrates in a spherical shell, asymptotically for large n. This in turn shows that the inequality 1 less than or equal to (K)integral \ x \(2)dx ((K)integral 1/\ x \ dx)(2) is close to an equality for the mentioned class of isotropic convex sets, a symptotically for large n. It also implies a 'central limit property' for t his class.