A probabilistic approach to the geometry of the ℓᵨⁿ-ball

Abstract

This article investigates, by probabilistic methods, various geometric questions on Bᵨⁿ, the unit ball of ℓᵨⁿ. We propose realizations in terms of independent random variables of several distributions on Bᵨⁿ, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in Bᵨⁿ. As another application, we compute moments of linear functionals on Bᵨⁿ, which gives sharp constants in Khinchine’s inequalities on Bᵨⁿ and determines the ψ₂-constant of all directions on Bᵨⁿ. We also study the extremal values of several Gaussian averages on sections of Bᵨⁿ (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.