A probabilistic approach to Lorentz balls

Abstract

We develop a probabilistic approach to study the volumetric and geometric properties of unit balls ${\mathbb{B}}_{q,1}^n$ of finite-dimensional Lorentz sequences spaces ${\ell }_{q,1}^n$. More precisely, we show that the empirical distribution of a random vector $X^{(n)}$ uniformly distributed on the volume normalized Lorentz ball in ${\mathbb{R}}^n$ 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 $k\in \mathbb{N}$ of coordinates of $X^{(n)}$ as $n\to \infty$. 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 ${\ell }_q^n$ 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 ${\ell }_p^n$ balls.

27 pages, 1 figure

References in corpus (1)

• A probabilistic approach to the geometry of the \ell_p^n-ball