@article{KabluchkoProchnoSonnleitner2024, author = {Kabluchko, Zakhar and Prochno, Joscha and Sonnleitner, Mathias}, title = {A probabilistic approach to Lorentz balls l(^n)(q,1)}, series = {Journal of Functional Analysis (Online ISSN: 1096-0783)}, volume = {2025}, journal = {Journal of Functional Analysis (Online ISSN: 1096-0783)}, number = {288, 1}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jfa.2024.110682}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18878}, pages = {32 Seiten}, year = {2024}, abstract = {We develop a probabilistic approach to study the volumetric and geometric properties of unit balls |B(^n)(q,1) of finite-dimensional Lorentz sequence spaces l(^n)(q,1). More precisely, we show that the empirical distribution of a random vector X^(n) 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 Poincar{\´e}-Maxwell-Borel principle for any fixed number k in |N of coordinates of X^(n) as n grows infinitly. 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 l(^n)(q) balls, where a Gumbel distribution appears in the limit. Finally, we prove a Schechtman-Schmuckenschl{\"a}ger type result for the asymptotic volume of intersections of volume normalized l(^n)(q,1) and l(^n)(p) balls.}, language = {en} }