A probabilistic approach to the geometry of the ℓpn-ball

Abstract

This article investigates, by probabilistic methods, various geometric questions on B_pⁿ, the unit ball of ℓ_pⁿ. We propose realizations in terms of independent random variables of several distributions on B_pⁿ, 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_pⁿ. As another application, we compute moments of linear functionals on B_pⁿ, which gives sharp constants in Khinchine’s inequalities on B_pⁿ and determines the ψ₂-constant of all directions on B_pⁿ. We also study the extremal values of several Gaussian averages on sections of B_pⁿ (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.