A probabilistic approach to the geometry of the \ell_p^n-ball

Details A probabilistic approach to the geometry of the \ell_p^n-ball A probabilistic approach to the geometry of the \ell_p^n-ball

2005-03-29

arxiv.org/abs/math/0503650v1

Annals of Probability 2005, Vol. 33, No. 2, 480-513

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000874 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000874

This article investigates, by probabilistic methods, various geometric questions on B_p^n, the unit ball of \ell_p^n. We propose realizations in terms of independent random variables of several distributions on B_p^n, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B_p^n. As another application, we compute moments of linear functionals on B_p^n, which gives sharp constants in Khinchine's inequalities on B_p^n and determines the \psi_2-constant of all directions on B_p^n. We also study the extremal values of several Gaussian averages on sections of B_p^n (including mean width and \ell-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in \ell_2 and to covering numbers of polyhedra complete the exposition.

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