Vol. 100, No. 1, 1982

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Martingale proofs of some geometrical results in Banach space theory

Kenneth Kunen and Haskell Paul Rosenthal

Vol. 100 (1982), No. 1, 153–175

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 “𝜀-strong extreme points” and the approximate Krein-Milman property are introduced, and the intimate connections between these notions and “δ-trees” are explored. A self-contained treatment is given of the necessary martingale preliminaries, phrased in terms of quasi-martingales.

Mathematical Subject Classification 2000
Primary: 46B20
Secondary: 60B11, 60G42
Received: 25 May 1981
Published: 1 May 1982
Kenneth Kunen
Haskell Paul Rosenthal