Metric entropy of convex hulls in type \(p\) spaces--the critical case (Q2759022)

Let \(A\) be a precompact subset of a Banach space \(E\) of type \(p \in (1,2]\). For each \(x\in E\) the closed ball of radius \(\varepsilon>0\) around \(x\) is denoted by \(B(x,\varepsilon)\). Given a number \(n\in\mathbb{N}\) the numbers NEWLINE\[NEWLINE\varepsilon_n(A): =\inf\bigl\{ \varepsilon>0: \exists x_1,\dots, x_n\in A\text{ such that }A\subseteq B(x_1,\varepsilon) \cup\cdots\cup B(x_n, \varepsilon) \bigr\}NEWLINE\]NEWLINE and \(e_n(A): =\varepsilon_{2^{n-1}}(A)\) are called the \(n\)th entropy number and the \(n\)th dyadic entropy number of \(A\), respectively. Continuing the investigations by \textit{B. Carl}, \textit{I. Kyrezi} and \textit{A. Pajor} [J. Lond. Math. Soc., II. Ser. 60, 871-896 (1991; Zbl 0976.46009)]; the authors aim to give an upper estimate of the dyadic entropy numbers of the absolutely convex hull aco \(A\) of \(A\) in terms of the dyadic entropy numbers of \(A\). If \(\alpha=1-1/p\), \(\beta\in (-\infty,1]\) and \(f:[1,\infty) \to(0,\infty)\) is an increasing function such that \(e_n(A)\leq cn^{-\alpha} (\log(n+1))^{-\beta} f(n)\) for all \(n \in \mathbb{N}\), then they prove that \(e_n(\text{eco} A)\leq cn^{-\alpha} (\log(n+ 1))^{-\beta+1} f(n)\) for every \(n\in\mathbb{N}\), where \(c>0\) is a suitable constant independent of \(n\). Further, it is shown that this estimate of \(e_n (\text{eco} A)\) is asymptotically optimal under natural conditions on \(E\) and \(f\).