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

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 $n \in \mathbb{N}$

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.