  We study the empirical process indexed by F^2=\{f^2 : f \in F\}, where F is a
class of mean-zero functions on a probability space. We present a sharp bound
on the supremum of that process which depends on the \psi_1 diameter of the
class F (rather than on the \psi_2 one) and on the complexity parameter
\gamma_2(F,\psi_2). In addition, we present optimal bounds on the random
diameters \sup_{f \in F} \max_{|I|=m} (\sum_{i \in I} f^2(X_i))^{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.
