We study the empirical process supf∈F |N −1 ∑N i=1 f 2 (Xi) − Ef 2 |, where F is a class of mean-zero functions on a probability space (Ω, µ) and (Xi) N i=1 are selected independently according to µ. We present a sharp bound on this supremum that depends on the ψ1 diameter of the class F (rather than on the ψ2 one) and on the complexity parameter γ2(F, ψ2). In addition, we present optimal bounds on the random diameters sup f∈F max |I|=m ( ∑ i∈I f 2 (Xi)) 1/2 using the same parameters. As applications, we extend several well known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on R n.