The Annals of Probability
- Ann. Probab.
- Volume 33, Number 2 (2005), 480-513.
A probabilistic approach to the geometry of the ℓpn-ball
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.
Ann. Probab., Volume 33, Number 2 (2005), 480-513.
First available in Project Euclid: 3 March 2005
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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