Martingale proofs of some geometrical results in Banach space theory
Let X be a Banach space. A sequence \(\{f_ n\}\) of X-valued functions belonging to \(L^ p(X)\), 1\(\leq p\leq \infty\), is called a p-quasi- martingale with respect to a sequence of \(\sigma\)-subalgebras (\({\mathcal A}_ n)\) with \({\mathcal A}_ n\subset {\mathcal A}_{n+1}\), if each \(f_ n\) is \({\mathcal A}_ n\)-measurable and \(\sum^{\infty}_{n=0}\| {\mathcal E}_ nf_{n+1}-f_ n\|<\infty,\) where \({\mathcal E}_ n\) denotes the conditional expectation with respect to \({\mathcal A}_ n\). Such a generalization of the notion of martingale is used to prove standard results involving dentability, strongly exposed points and the Radon- Nikodym Property (RNP). The arguments reveal the power of the explicit use of quasi-martingales and also set up techniques for further results. In the final section, the concept of an ''\(\epsilon\)-extreme point'' is introduced, namely, if K is a closed, bounded convex set and \(\epsilon>0\), then \(x\in K\) is an \(\epsilon\)-strong extreme point of K if there is a \(\delta>0\) so that \(k_ 1,k_ 2\in K\) satisfy \(\|(k_ 1+k_ 2)/2-x\|<\delta,\) then \(\| k_ 1-k_ 2\|<2\epsilon.\) It is shown that if K has no \(\epsilon\)-strong extreme point, then, for any \(\delta<\epsilon\), K contains \(\delta\)-trees, i.e., dyadic martingales with differences everywhere \(\delta\)-bounded away from zero. An open problem is whether these conditions imply that such a set K is the closed convex hull of its \(\epsilon\)-strong extreme points for every \(\epsilon>0\). If this were not the case, it is shown K admits a dyadic K- valued martingale with differences \(\epsilon\)-bounded away from zero on a set of positive measure. This result implies that for known examples of Banach spaces which fail RNP but yet do not contain bounded trees, every closed bounded convex subset of K is indeed the closed, convex hull of its set of \(\epsilon\)-strong extreme points for every \(\epsilon>0\).
scientific article; zbMATH DE number 3849853