The Annals of Probability

A probabilistic approach to the geometry of the ℓpn-ball

Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor

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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: 3 March 2005

Permanent link to this document
http://projecteuclid.org/euclid.aop/1109868589

Digital Object Identifier
doi:10.1214/009117904000000874

Mathematical Reviews number (MathSciNet)
MR2123199

Subjects
Primary: 60E15: Inequalities; stochastic orderings 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52A38: Length, area, volume [See also 26B15, 28A75, 49Q20] 52A40: Inequalities and extremum problems

Keywords
ℓ_p^n-ball Gaussian measure extremal sections

Citation

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


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