This is an announcement for the paper "On the metric entropy of the Banach-Mazur compactum" by Gilles Pisier.

Abstract: We study of the metric entropy of the metric space $\cl B_n$ of all $n$-dimensional Banach spaces (the so-called Banach-Mazur compactum) equipped with the Banach-Mazur (multiplicative) ``distance" $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({\cl B_n},d, 1+\vp)$ is the smallest number of ``balls" of ``radius" $1+\vp$ that cover $\cl B_n$, then for any $\vp>0$ we have $$0<\liminf_{n\to \infty} \log\log N(\cl B_n,d,1+\vp)\le \limsup_{n\to \infty} \log\log N(\cl B_n,d,1+\vp)<\infty.$$ We also prove similar results for the matricial operator space analogues.

Archive classification: math.FA

Submitted from: pisier@math.jussieu.fr

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1306.5325

or

http://arXiv.org/abs/1306.5325