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

by    J. Creutzig, I. Steinwart

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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