2020/11/15 by David Alonso–Gutiérrez, Joscha Prochno, Alonso-Gutiérrez, David +1
Mathematics · #46B45 #52A23 #60F05 #60F10 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #Primary 46B06 #Probability (math.PR) #Secondary 46B09
paper · pdf · doi:10.48550/arxiv.2011.07523
openalex publication_date 2020/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the asymptotic thin-shell width concentration for\nrandom vectors uniformly distributed in Orlicz balls. We provide both\nasymptotic upper and lower bounds on the probability of such a random vector\nXn being in a thin shell of radius \√(n) times the asymptotic value of\nn-1/2\( mathbb E\[\‖ Xn\‖22\]\)1/2 (as\nn\→\∞), showing that in certain ranges our estimates are optimal. In\nparticular, our estimates significantly improve upon the currently best known\ngeneral Lee-Vempala bound when the deviation parameter t=tn goes down to\nzero as the dimension n of the ambient space increases. We shall also\ndetermine in this work the precise asymptotic value of the isotropic constant\nfor Orlicz balls. Our approach is based on moderate deviation principles and a\nconnection between the uniform distribution on Orlicz balls and Gibbs measures\nat certain critical inverse temperatures with potentials given by Orlicz\nfunctions, an idea recently presented by Kabluchko and Prochno in [The maximum\nentropy principle and volumetric properties of Orlicz balls, J. Math. Anal.\nAppl. bf 495(1) 2021, 1--19].\n