  Let $F$ be a class of functions on a probability space $(\Omega,\mu)$ and let
$X_1,...,X_k$ be independent random variables distributed according to $\mu$.
We establish high probability tail estimates of the form $\sup_{f \in F} |\{i :
|f(X_i)| \geq t \}$ using a natural parameter associated with $F$. We use this
result to analyze weakly bounded empirical processes indexed by $F$ and
processes of the form $Z_f=|k^{-1}\sum_{i=1}^k |f|^p(X_i)-\E|f|^p|$ for $p>1$.
We also present some geometric applications of this approach, based on
properties of the random operator $\Gamma=k^{-1/2}\sum_{i=1}^k
\inr{X_i,\cdot}e_i$, where the $(X_i)_{i=1}^k$ are sampled according to an
isotropic, log-concave measure on $\R^n$.
