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Geometry of \(\ell _p^n\,\text{-Balls}\): Classical Results and Recent Developments

  • Joscha ProchnoEmail author
  • Christoph Thäle
  • Nicola Turchi
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)

Abstract

In this article we first review some by-now classical results about the geometry of p-balls \(\mathbb {B}_p^n\) in \(\mathbb {R}^n\) 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 \(\mathbb {B}_p^n\). We discuss their relation to the classical results and give hints to various extensions that are available in the existing literature.

Keywords

Asymptotic geometric analysis \(\ell _p^n\text{-Balls}\) Central limit theorem Law of large numbers Large deviations Polar integration formula 

2010 Mathematics Subject Classification

46B06 47B10 60B20 60F10 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Joscha Prochno
    • 1
    Email author
  • Christoph Thäle
    • 2
  • Nicola Turchi
    • 2
  1. 1.Institut für Mathematik und Wissenschaftliches RechnenKarl-Franzens-Universität GrazGrazAustria
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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