A probabilistic approach to the geometry of the l p n-ball

Franck Barthe, Olivier Guédon, Shahar Mendelson, Assaf Naor

Research output: Contribution to journalArticle

Abstract

This article investigates, by probabilistic methods, various geometric questions on B p n, the unit ball of l p n. We propose realizations in terms of independent random variables of several distributions on B p n, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B p n. As another application, we compute moments of linear functional on B p n, which gives sharp constants in Khinchine's inequalities on B p n and determines the ψ 2-constant of all directions on B p n. We also study the extremal values of several Gaussian averages on sections of B p n (including mean width and l-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition.

Original languageEnglish (US)
Pages (from-to)480-513
Number of pages34
JournalAnnals of Probability
Volume33
Issue number2
DOIs
StatePublished - Mar 2005

Fingerprint

Probabilistic Approach
Ball
Sharp Constants
Covering number
Probabilistic Methods
Linear Functional
Independent Random Variables
Unit ball
Polyhedron
Balancing
Monotonicity
Vary
Moment
Norm
Geometry
Independence
Probabilistic methods
Random variables

Keywords

  • Extremal sections
  • Gaussian measure
  • L -ball

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability

Cite this

A probabilistic approach to the geometry of the l p n-ball. / Barthe, Franck; Guédon, Olivier; Mendelson, Shahar; Naor, Assaf.

In: Annals of Probability, Vol. 33, No. 2, 03.2005, p. 480-513.

Research output: Contribution to journalArticle

Barthe, F, Guédon, O, Mendelson, S & Naor, A 2005, 'A probabilistic approach to the geometry of the l p n-ball', Annals of Probability, vol. 33, no. 2, pp. 480-513. https://doi.org/10.1214/009117904000000874
Barthe, Franck ; Guédon, Olivier ; Mendelson, Shahar ; Naor, Assaf. / A probabilistic approach to the geometry of the l p n-ball. In: Annals of Probability. 2005 ; Vol. 33, No. 2. pp. 480-513.
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