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

113 Scopus citations

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 1 2005
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Extremal sections
  • Gaussian measure
  • L -ball

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