Metric entropy of convex hulls in type $p$ spaces—The critical case

Abstract:

Given a precompact subset $A$ of a type $p$ Banach space $E$, where $p\in (1,2]$, we prove that for every $\beta \in [0,1)$ and all $n \in \mathbb {N}$ \begin{equation*} \sup _{k \le n}k^{1/p’} (\log k)^{\beta -1} e_k(\mathrm {aco } A) \ \le \ c \ \sup _{k \le n}k^{1/p’} (\log k)^{\beta }e_k(A) \end{equation*} holds, where $\mathrm {aco } A$ is the absolutely convex hull of $A$ and $e_k(.)$ denotes the $k^{th}$ dyadic entropy number. With this inequality we show in particular that for given $A$ and $\beta \in ( -\infty , 1)$ with $e_n(A) \le n^{-1/p’} (\log n)^{-\beta }$ for all $n \in \mathbb {N}$ the inequality $e_n(\mathrm {aco } A) \le c\ n^{-1/p’}(\log n)^{-\beta + 1}$ holds true for all $n \in \mathbb {N}$. We also prove that this estimate is asymptotically optimal whenever $E$ has no better type than $p$. For $\beta =0$ this answers a question raised by Carl, Kyrezi, and Pajor which has been solved up to now only for the Hilbert space case by F. Gao.