Majorizing measures and proportional subsets of bounded orthonormal systems
Résumé
In this article we prove that for any orthonormal system (φj)nj=1 ⊂ L2 that is bounded in L∞, and any 1 < k < n, there exists a subset I of cardinality greater than n − k such that on span{φi}i∈I, the L1 norm and the L2 norm are equivalent up to a factor μ(logμ)5/2, where μ = ﰅn/k√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.