This is an announcement for the paper "Majorizing measures and proportional subsets of bounded orthonormal systems" by Olivier Guedon, Shahar Mendelson, Alain Pajor, and Nicole Tomczak-Jaegermann.
Abstract: In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\spa{\vphi_i}_{i \in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\mu (\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.
Archive classification: math.FA math.PR
The source file(s), arXiv.tex: 50357 bytes, is(are) stored in gzipped form as 0801.3556.gz with size 16kb. The corresponding postcript file has gzipped size 130kb.
Submitted from: alain.pajor@univ-mlv.fr
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