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

Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor
Source: Ann. Probab. Volume 33, Number 2 (2005), 480-513.

#### 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.

First Page:
Primary Subjects: 60E15, 52A20, 52A38, 52A40
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aop/1109868589
Digital Object Identifier: doi:10.1214/009117904000000874
Mathematical Reviews number (MathSciNet): MR2123199

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