Entropy, dimension and the Elton-Pajor Theorem

TL;DR

This paper explores how the VC dimension of convex bodies in R^n influences their entropy, leading to optimal results in Elton's theorem and a uniform central limit theorem for real-valued functions.

Contribution

It establishes a direct link between VC dimension and entropy of convex bodies, providing new optimal bounds and theoretical insights.

Findings

VC dimension governs the entropy of convex bodies

Results include the optimal Elton's theorem

Proves a uniform central limit theorem in the real-valued case

Abstract

The Vapnik-Chervonenkis dimension of a set K in R^n is the maximal dimension of the coordinate cube of a given size, which can be found in coordinate projections of K. We show that the VC dimension of a convex body governs its entropy. This has a number of consequences, including the optimal Elton's theorem and a uniform central limit theorem in the real valued case.