Constructions of Majorizing Measures, Bernoulli processes and cotype

Details Constructions of Majorizing Measures, Bernoulli processes and cotype Constructions of Majorizing Measures, Bernoulli processes and cotype

1994-06-07

arxiv.org/abs/math/9406216v1

Mathematics

Functional Analysis

Scientific paper

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 $N(A,\varepsilon B)$ for all $\varepsilon>0$ 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.

Affiliated with

Also associated with

No associations

Rating

Constructions of Majorizing Measures, Bernoulli processes and cotype does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Constructions of Majorizing Measures, Bernoulli processes and cotype, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Constructions of Majorizing Measures, Bernoulli processes and cotype will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-421562