Entropy and the Combinatorial Dimension
arXiv:math/0203275
Abstract
We solve Talagrand's entropy problem: the L_2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley's theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton's Theorem and estimates on the uniform central limit theorem in the real valued case.
A final version of the paper (Inventiones Math., to appear) Only two applications added: one to Asymptotic Geometry (the optimal Elton Theorem) and the other to empirical processes (the uniform central limit theorem)