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)


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.


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 


  1. 1.
    D. Alonso-Gutiérrez, J. Prochno, C. Thäle, Large deviations for high-dimensional random projections of \(\ell _{p}^{n}\)-balls. Adv. Appl. Math. 99, 1–35 (2018)Google Scholar
  2. 2.
    M. Anttila, K. Ball, I. Perissinaki, The central limit problem for convex bodies. English. Trans. Am. Math. Soc. 355(12), 4723–4735 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Artstein-Avidan, A. Giannopoulos, V. Milman, Asymptotic Geometric Analysis. Part I. Mathematical Surveys and Monographs, vol. 202 (American Mathematical Society, Providence, 2015), pp. xx+451Google Scholar
  4. 4.
    F. Barthe, O. Guédon, S. Mendelson, A. Naor, A probabilistic approach to the geometry of the \(l_{p}^{n}\)-ball. Ann. Probab. 33(2), 480–513 (2005)Google Scholar
  5. 5.
    S. Brazitikos, A. Giannopoulos, P. Valettas, B.-H. Vritsiou, Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs, vol. 196 (American Mathematical Society, Providence, 2014), pp. xx+594Google Scholar
  6. 6.
    A. Dembo, O. Zeitouni, Large Deviations. Techniques and Applications. Stochastic Modelling and Applied Probability, vol. 38. Corrected reprint of the second (1998) edition (Springer, Berlin, 2010), pp. xvi+396Google Scholar
  7. 7.
    F. den Hollander, Large Deviations. Fields Institute Monographs, vol. 14 (American Mathematical Society, Providence, 2000), pp. x+143Google Scholar
  8. 8.
    P. Dirichlet, Sur une nouvelle méthode pour la détermination des intégrales multiples. J. Math. Pures Appl. 4, 164–168 (1839)Google Scholar
  9. 9.
    N. Gantert, S. Kim, K. Ramanan, Large deviations for random projections of p balls. Ann. Probab. 45(6B), 4419–4476 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Z. Kabluchko, J. Prochno, C. Thäle, Exact asymptotic volume and volume ratio of Schatten unit balls. ArXiv e-prints (Apr. 2018). arXiv: 1804.03467 [math.FA]Google Scholar
  11. 11.
    Z. Kabluchko, J. Prochno, C. Thäle, Intersection of unit balls in classical matrix ensembles. Israel J. Math. (to appear). ArXiv e-prints (Apr. 2018). arXiv: 1804.03466 [math.FA]Google Scholar
  12. 12.
    Z. Kabluchko, J. Prochno, C. Thäle, Sanov-type large deviations in Schatten classes. Ann. Inst. H. Poincaré Probab. Statist. (to appear). ArXiv e-prints (Aug. 2018). arXiv: 1808.04862 [math.PR]Google Scholar
  13. 13.
    Z. Kabluchko, J. Prochno, C. Thäle, High-dimensional limit theorems for random vectors in \(\ell _{p}^{n}\)-balls. Commun. Contemp. Math. 21, 1750092 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    O. Kallenberg, Foundations of Modern Probability, Second. Probability and Its Applications (Springer, New York, 2002), pp. xx+638CrossRefGoogle Scholar
  15. 15.
    S. Kim, Problems at the interface of probability and convex geometry: random projections and constrained processes. Ph.D. thesis. Brown University, 2017Google Scholar
  16. 16.
    S. Kim, K. Ramanan, A conditional limit theorem for high-dimensional p spheres. ArXiv e-prints (Sept. 2015). arXiv: 1509.05442 [math.PR]Google Scholar
  17. 17.
    A. Naor, The surface measure and cone measure on the sphere of \(\ell _{p}^{n}\). Trans. Am. Math. Soc. 359(3), 1045–1079 (2007)Google Scholar
  18. 18.
    A. Naor, D. Romik, Projecting the surface measure of the sphere of \(\ell _{p}^{n}\). English. Ann. Inst. Henri Poincaré, Probab. Stat. 39(2), 241–261 (2003)Google Scholar
  19. 19.
    G. Pisier, The Volume of Convex Bodies and Banach Space Geometry (Cambridge University Press, Cambridge, 1999)zbMATHGoogle Scholar
  20. 20.
    S. Rachev, L. Rüschendorf, Approximate independence of distributions on spheres and their stability properties. Ann. Probab. 19(3), 1311–1337 (1991)MathSciNetCrossRefGoogle Scholar
  21. 21.
    G. Schechtman, M. Schmuckenschläger, Another remark on the volume of the inter-section of two \(l_{p}^{n}\) balls, in Geometric Aspects of Functional Analysis (1989–90), vol. 1469. Lecture Notes in Mathematics (Springer, Berlin, 1991), pp. 174–178Google Scholar
  22. 22.
    G. Schechtman, J. Zinn, On the volume of the intersection of two \(l_{p}^{n}\)balls. Proc. Am. Math. Soc. 110(1), 217–224 (1990)Google Scholar
  23. 23.
    M. Schmuckenschläger, Volume of intersections and sections of the unit ball of \(\ell _{p}^{n}\). Proc. Am. Math. Soc. 126(5), 1527–1530 (1998)Google Scholar
  24. 24.
    M. Schmuckenschläger, CLT and the volume of intersections of \(\ell _{p}^{n}\)-balls. English. Geom. Dedicata 85(1–3), 189–195 (2001)Google Scholar
  25. 25.
    D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. I. Commun. Math. Phys. 155(1), 71–92 (1993)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations