A geometria de L(mRn) e aplicações

Abstract

In this work, we deal with the geometry of L(mRn) and exhibit a characterization, obtained in the paper [7], for the extreme points of the closed unit ball in this space, which reveals an algorithm that gives these points through finitely many elementary steps ([7], Section 4). These results are used, together with the Minkowski (Krein- Milman) Theorem, to put in practice a new approach to the problem of finding the optimal constants for the Bohnenblust-Hille inequality ([7], Section 5.1). Towards this end, with the aim to refute or corroborate the conjecture that the optimal constant is in fact 21−1/m (see [18]), we implement some versions of the algorithm into the software Mathematica. We realize that there exists extremal points in BL(3R3) with up to 22 monomials and the conjecture was corroborated with these discoveries. The same approach applied to the Grothendieck inequality is discussed.