  We evaluate the shattering dimension of various classes of linear functionals
on various symmetric convex sets. The proofs here relay mostly on methods from
the local theory of normed spaces and include volume estimates, factorization
techniques and tail estimates of norms, viewed as random variables on Euclidean
spheres. The estimates of shattering dimensions can be applied to obtain error
bounds for certain classes of functions, a fact which was the original
motivation of this study. Although this can probably be done in a more
traditional manner, we also use the approach presented here to determine
whether several classes of linear functionals satisfy the uniform law of large
numbers and the uniform central limit theorem.
