A probabilistic approach to the geometry of the l _p ⁿ-ball

This article investigates, by probabilistic methods, various geometric questions on B _p ⁿ, the unit ball of l _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 functional 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 l-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in l ₂ and to covering numbers of polyhedra complete the exposition.