Moment inequalities and central limit properties of isotropic convex bodies (Q697353)

The authors investigate isotropic convex bodies \(K\subseteq \mathbb R^n\) , centred at the origin and normed to volume one, in arbitrary dimensions. It is shown that a certain subset of these bodies, specified by bounds on the second and fourth moments, is invariant under forming `expanded joins'. Considering such a body \(K\) as a probability space and taking \(u\in S^{n-1}\), one can define the 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 \(u\in S^{n-1}\). The authors 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.