# High Dimensional Probability VIIIGeometry of ℓpn-Balls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\ell _p^n\,\text{-Balls}$$ \end{document}: Classical Results and Recent Developments

High Dimensional Probability VIII: Geometry of ℓpn-Balls\documentclass[12pt]{minimal}... [In this article we first review some by-now classical results about the geometry of ℓp-balls 𝔹pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {B}_p^n$$ \end{document} in ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {R}^n$$ \end{document} and provide modern probabilistic arguments for them. We also present some more recent developments including a central limit theorem and a large deviations principle for the q-norm of a random point in 𝔹pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {B}_p^n$$ \end{document}. We discuss their relation to the classical results and give hints to various extensions that are available in the existing literature.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# High Dimensional Probability VIIIGeometry of ℓpn-Balls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\ell _p^n\,\text{-Balls}$$ \end{document}: Classical Results and Recent Developments

Part of the Progress in Probability Book Series (volume 74)
Editors: Gozlan, Nathael; Latała, Rafał; Lounici, Karim; Madiman, Mokshay
29 pages

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# References (14)

Publisher
Springer International Publishing
Copyright
© Springer Nature Switzerland AG 2019
ISBN
978-3-030-26390-4
Pages
121 –150
DOI
10.1007/978-3-030-26391-1_9
Publisher site
See Chapter on Publisher Site

### Abstract

[In this article we first review some by-now classical results about the geometry of ℓp-balls 𝔹pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {B}_p^n$$ \end{document} in ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {R}^n$$ \end{document} and provide modern probabilistic arguments for them. We also present some more recent developments including a central limit theorem and a large deviations principle for the q-norm of a random point in 𝔹pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {B}_p^n$$ \end{document}. We discuss their relation to the classical results and give hints to various extensions that are available in the existing literature.]

Published: Nov 27, 2019

Keywords: Asymptotic geometric analysis; ℓpn-Balls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\ell _p^n\text{-Balls}$$ \end{document}; Central limit theorem; Law of large numbers; Large deviations; Polar integration formula; 46B06; 47B10; 60B20; 60F10

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