Metric Entropy of Convex Hulls in Type $p$ spaces -- the Critical Case

Preprint series: 99-58, Report

MSC:
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
46B07 Local theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Abstract: Given a precompact subset $A$ of a type $p$ Banach space $E$, $1< p \le 2$,
we prove that for every $0\leq \beta <1$ it holds
\begin{equation*}
\sup_{k \le n}k^{1/p'} (\log k)^{\beta -1} e_k(\aco A)
\ \le C \ \sup_{k \le n}k^{1/p'} (\log k)^{\beta}e_k(A),
\end{equation*}
where $\aco A$ 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 $e_n(A) \preceq n^{-1/p'} (\log n)^{-\beta}$
implies $e_n(\aco A) \preceq n^{-1/p'}(\log n)^{-\beta + 1}$ for all $-\infty <\beta<1$.
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 in \cite{CKP} which has been answered up to now only for the Hilbert
space case in \cite{Ga}.

Keywords: Metric entropy, entropy numbers, convex sets