### 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 language | English (US) |
---|---|

Pages (from-to) | 480-513 |

Number of pages | 34 |

Journal | Annals of Probability |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2005 |

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### Keywords

- Extremal sections
- Gaussian measure
- L -ball

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*33*(2), 480-513. https://doi.org/10.1214/009117904000000874

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

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 33, no. 2, pp. 480-513. https://doi.org/10.1214/009117904000000874

}

TY - JOUR

T1 - A probabilistic approach to the geometry of the l p n-ball

AU - Barthe, Franck

AU - Guédon, Olivier

AU - Mendelson, Shahar

AU - Naor, Assaf

PY - 2005/3

Y1 - 2005/3

N2 - 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.

AB - 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.

KW - Extremal sections

KW - Gaussian measure

KW - L -ball

UR - http://www.scopus.com/inward/record.url?scp=17044432290&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17044432290&partnerID=8YFLogxK

U2 - 10.1214/009117904000000874

DO - 10.1214/009117904000000874

M3 - Article

AN - SCOPUS:17044432290

VL - 33

SP - 480

EP - 513

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -