17 - Probabilistic Tools II

Summary

We continue the study initiated in Chapter 7 of polynomials with small norms. This time the norm of the polynomial is not taken as the supremum on the n-dimensional polydisc, we take it on B_X, the unit ball of some Banach space. The goal is to show that, given a polynomial, signs can be found in such a way that the norm of the new polynomial, whose coefficients are the original ones multiplied by the signs, has small norm. We do this with three different approaches. The first two approaches use Rademacher random variables as the main probabilistic tools. The first one is based on finding out how many balls of a fixed radius are needed to cover B_X while the second one uses entropy integrals and a good estimate for the entropy numbers of the inclusions between l_p spaces. The third approach is different, and relies on Gaussian random variables, Slepian’s lemma and the fact that Rademacher averages are dominated by Gaussian averages. This approach also allows to get estimates for vector-valued polynomials.