Entropy numbers of finite-dimensional embeddings

Abstract

Entropy numbers and covering numbers of sets and operators are well known geometric notions, which found many applications in various fields of mathematics, statistics, and computer science. Their values for finite-dimensional embeddings $id:\ell_p^n\to \ell_q^n$, $0<p,q\le\infty$, are known (up to multiplicative constants) since the pioneering work of Sch\"utt in 1984, with later improvements by Edmunds and Triebel, K\"uhn and Gu\'edon and Litvak. The aim of this survey is to give a self-contained presentation of the result and an overview of the different techniques used in its proof.