### 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) |
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Pages (from-to) | 480-513 |

Number of pages | 34 |

Journal | Annals of Probability |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2005 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Extremal sections
- Gaussian measure
- L -ball

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## Cite this

_{p}

^{n}-ball.

*Annals of Probability*,

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