# Combinatorics of random processes and sections of convex bodies

### Abstract

We find a sharp combinatorial bound for the metric entropy of sets in $\mathbb{R}^n$ and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in $\mathbb{R}^n$. In the operator theory, this essentially proves for all normed spaces the restricted invertibility principle of Bourgain and Tzafriri.

## Authors

Mark Rudelson

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States

Roman Vershynin

Department of Mathematics, University of California, Davis, CA 95616, United States