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

Abstract

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 ψ_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.

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

Cited by in corpus (12)

• The Logarithmic Minkowski Problem
• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• The dual Minkowski problem for negative indices
• An isoperimetric inequality on the ${\ell }_p$ balls
• Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling
• Threshold condensation to singular support for a Riesz equilibrium problem
• Slicing ${\ell }_p$-balls reloaded: stability, planar sections in ${\ell }_1$
• High-dimensional limit theorems for random vectors in ${\ell }_p^n$-balls
• Spectral flatness and the volume of intersections of $p$-ellipsoids
• Discrepancy, chaining and subgaussian processes
• Sharp Asymptotics for $q$-Norms of Random Vectors in High-Dimensional ${\ell }_p^n$-Balls
• A probabilistic approach to Lorentz balls