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


Authors: Jakob Creutzig and Ingo Steinwart
Journal: Proc. Amer. Math. Soc. 130 (2002), 733-743
MSC (2000): Primary 41A46; Secondary 46B07, 46B20, 47B37, 52A07
DOI: https://doi.org/10.1090/S0002-9939-01-06256-6
Published electronically: August 28, 2001
MathSciNet review: 1866028
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Abstract: 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} $

\begin{displaymath}\sup_{k \le n}k^{1/p'} (\log k)^{\beta -1} e_k(\mathrm{aco\, } A) \le c \sup_{k \le n}k^{1/p'} (\log k)^{\beta}e_k(A) \end{displaymath}

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.


References [Enhancements On Off] (What's this?)

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Additional Information

Jakob Creutzig
Affiliation: FSU Jena, Ernst–Abbe–Platz 1-4, 07743 Jena, Germany
Email: jakob@creutzig.de

Ingo Steinwart
Affiliation: FSU Jena, Ernst–Abbe–Platz 1-4, 07743 Jena, Germany
Email: steinwart@minet.uni-jena.de

DOI: https://doi.org/10.1090/S0002-9939-01-06256-6
Keywords: Metric entropy, entropy numbers, convex sets
Received by editor(s): April 18, 2000
Received by editor(s) in revised form: September 6, 2000
Published electronically: August 28, 2001
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2001 American Mathematical Society

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