Constructions of Majorizing Measures, Bernoulli Processes and Cotype

We present three methods to construct majorizing measures in various settings. These methods are based on direct constructions of increasing sequences of partitions through a simple exhaustion procedure rather than on the construction of well separated ultrametric subspaces. The first scheme of construction provides a simple unified proof of the Majorizing Measure Theorem for Gaussian processes and of the following fact. If A,B are balanced convex sets in a vector space, and if A is sufficiently convex, a control of the covering numbers N(A, εB) for all ε > 0 implies the (a priori stronger) existence of a majorizing measure on A provided with the distance induced by B. This establishes, apparently for the first time, a clear link between geometry and majorizing measures, and generalizes the earlier results on majorizing measures on ellipsoids in Hilbert space, that were obtained by specific methods. Much of the rest of the paper is concerned with the structure of bounded Bernoulli (=Radmacher) processes. The main conjecture on their structure is reformulated in several ways, that are shown to be equivalent, and to be equivalent to the existence of certain majorizing measures. Two schemes of construction of majorizing measures related to this problem are presented. One allows to describe Bernoulli processes when the index set, provided with the supremum norm, is sufficiently small. The other allows to prove a weak form of the main conjecture. This result, while not sufficient to characterize boundedness of Bernoulli processes, allows to prove the remarkable fact that for any continuous operator T from C(K) to E, the Rademacher cotype-2 constant of T is controlled by the maximum of the Gaussian cotype-2 constant of T and of its (2, 1)-summing norm. It is also proved, as a consequence of one of the main inequalities on Bernoulli processes, that in a Banach space E of dimension n, at most n logn log log n vectors suffices to compute the Rademacher cotype 2 constant of E within a universal constant. 1 Introduction The notion of majorizing measure has allowed considerable progress in the study of certain stochastic processes, in particular Gaussian processes. Given a metric space (T, d), and a probability measure μ on T , we set γ1/2(T, d, μ) = sup x∈T ∫ ∞