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We find a sharp combinatorial bound for the metric entropy of sets in R^n andgeneral classes of functions. This solves two basic combinatorial conjectureson the empirical processes.  1. A class of functions satisfies the uniform Central Limit Theorem if thesquare root of its combinatorial dimension is integrable.  2. The uniform entropy is equivalent to the combinatorial dimension underminimal regularity.  Our method also constructs a nicely bounded coordinate section of a symmetricconvex body in R^n. In the operator theory, this essentially proves for allnormed spaces the restricted invertibility principle of Bourgain and Tzafriri.

Combinatorics of random processes and sections of convex bodies