Moment inequalities and central limit properties of isotropic convex bodies

Abstract

The object of our investigations are isotropic convex bodies \(K\subseteq \mathbb{R}^n\), centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies – specified by bounds on the second and fourth moments – is invariant under forming ‘expanded joinsrsquo;. Considering a body K as above as a probability space and taking \(u\in S^{n-1}\), we define random variables \(X_{K,u}=x\cdot u\) on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions \(u\in S^{n-1}\). We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with respect to \(L_1\)-convergence and \(L_\infty\)-convergence of the corresponding densities.