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

Abstract

It is shown that, for a certain subclass of istropic convex sets in ℝn, the mass concentrates in a spherical shell, asymptotically for largen. This in turn shows that the inequality $$1 \leqslant \int\limits_K {\left| x \right|^2 dx\left( {\int\limits_K {\frac{1}{{\left| x \right|}}dx} } \right)^2 } $$ is close to an equality for the mentioned class of isotropic convex sets, asymptotically for largen. It also implies a ‘central limit property’ for this class.