**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.)
- 52A07 Convex sets in topological vector spaces, See also {46A55}

**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*

**Upload:** 2000-05-05

**Update:** 2002-04-19

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