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March 2005 A probabilistic approach to the geometry of the ℓpn-ball
Franck Barthe, Olivier Guédon, Shahar Mendelson, Assaf Naor
Ann. Probab. 33(2): 480-513 (March 2005). DOI: 10.1214/009117904000000874

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.

Citation

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Franck Barthe. Olivier Guédon. Shahar Mendelson. Assaf Naor. "A probabilistic approach to the geometry of the ℓpn-ball." Ann. Probab. 33 (2) 480 - 513, March 2005. https://doi.org/10.1214/009117904000000874

Information

Published: March 2005
First available in Project Euclid: 3 March 2005

MathSciNet: MR2123199
Digital Object Identifier: 10.1214/009117904000000874

Subjects:
Primary: 52A20 , 52A38 , 52A40 , 60E15

Keywords: extremal sections , Gaussian measure , ℓ_p^n-ball

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 2 • March 2005
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