A probabilistic approach to Lorentz balls ℓq,1n

Abstract

We develop a probabilistic approach to study the volumetric and geometric properties of unit balls $B_{q,1}^n$ of finite-dimensional Lorentz sequence spaces ${\ell }_{q,1}^n$. More precisely, we show that the empirical distribution of a random vector $X^{\left(n\right)}$ uniformly distributed on its volume normalized unit ball 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 $k\in N$ of coordinates of $X^{\left(n\right)}$ as $n\to \infty$. Moreover, we prove a central limit theorem for the largest coordinate of $X^{\left(n\right)}$, demonstrating a quite different behavior than in the case of the ${\ell }_q^n$ balls, where a Gumbel distribution appears in the limit. Finally, we prove a Schechtman-Schmuckenschläger type result for the asymptotic volume of intersections of volume normalized ${\ell }_{q,1}^n$ and ${\ell }_p^n$ balls.