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- 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