Martingale proofs of some geometrical results in Banach space theory

Abstract

Martingale techniques are used to give a new proof of the theorem of J. Bourgain-R. R. Phelps that a closed bounded convex subset K of a Banach space is the closed convex hull of its set of strongly exposed points provided K has the Radon-Nikodym property. The new notions of "-strong extreme points" and the approximate Krein-Milman property are introduced, and the intimate connections between these notions and "d-trees" are explored. A self-contained treatment is given of the necessary martingale preliminaries, phrased in terms of quasi-martingales.