## The Annals of Probability

### A probabilistic approach to the geometry of the ℓpn-ball

#### Abstract

This article investigates, by probabilistic methods, various geometric questions on Bpn, the unit ball of ℓpn. We propose realizations in terms of independent random variables of several distributions on Bpn, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in Bpn. As another application, we compute moments of linear functionals on Bpn, which gives sharp constants in Khinchine’s inequalities on Bpn and determines the ψ2-constant of all directions on Bpn. We also study the extremal values of several Gaussian averages on sections of Bpn (including mean width and ℓ-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in ℓ2 and to covering numbers of polyhedra complete the exposition.

#### Article information

Source
Ann. Probab. Volume 33, Number 2 (2005), 480-513.

Dates
First available in Project Euclid: 3 March 2005

https://projecteuclid.org/euclid.aop/1109868589

Digital Object Identifier
doi:10.1214/009117904000000874

Mathematical Reviews number (MathSciNet)
MR2123199

#### Citation

Barthe, Franck; Guédon, Olivier; Mendelson, Shahar; Naor, Assaf. A probabilistic approach to the geometry of the ℓ p n -ball. Ann. Probab. 33 (2005), no. 2, 480--513. doi:10.1214/009117904000000874. https://projecteuclid.org/euclid.aop/1109868589.

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