On weakly bounded empirical processes

Let F be a class of functions on a probability space (Ω, μ) and let X ₁,...,X _k be independent random variables distributed according to μ. We establish an upper bound that holds with high probability on {\rm sup}_{f \in F} |\{i : |f(X_i)| \geq t \for every t > 0, and that depends on a natural geometric parameter associated with F. We use this result to analyze the supremum of empirical processes of the form Z_f = |k_-1∑_i=1^k|f|^p(X_i) - double-struck E sign|f|^p| for p > 1 using the geometry of F. We also present some geometric applications of this approach, based on properties of the random operator Γ = k{-1/2}\sum_{i=1}X _i , •e _i , where (X_i)_{i=1}k are sampled according to an isotropic, log-concave measure on {\mathbb R.