A probabilistic approach to Lorentz balls

Abstract

We develop a probabilistic approach to study the volumetric and geometric properties of unit balls B n q,1 of finite-dimensional Lorentz sequences spaces n q,1 . More precisely, we show that the empirical distribution of a random vector X (n) uniformly distributed on the volume normalized Lorentz ball in R n converges weakly to a compactly supported symmetric probability distribution with explicitly given density; as a consequence we obtain a weak Poincaré-Maxwell-Borel principle for any fixed number k ∈ N of coordinates of X (n) as n → ∞. Moreover, we prove a central limit theorem for the largest coordinate of X (n) , demonstrating a quite different behavior than in the case of the n q balls, where a Gumbel distribution appears in the limit. Last but not least, we prove a Schechtman-Schmuckenschläger type result for the asymptotic volume of intersections of volume normalized Lorentz and n p balls.