  We investigate measures of complexity of function classes based on continuity
moduli of Gaussian and Rademacher processes. For Gaussian processes, we obtain
bounds on the continuity modulus on the convex hull of a function class in
terms of the same quantity for the class itself. We also obtain new bounds on
generalization error in terms of localized Rademacher complexities. This allows
us to prove new results about generalization performance for convex hulls in
terms of characteristics of the base class. As a byproduct, we obtain a simple
proof of some of the known bounds on the entropy of convex hulls.
