The concepts of
(strong) martingale representations and coordinatizations are defined, and the
notion of a well-separated bush is crystallized. It is proved that if ℬ is a
well-separated uniformly bounded bush such that ℬ is a strong martingale
representation for its closed convex hull W, then W contains no extreme
points. It is moreover proved that if K is a closed bounded convex subset of a
Banach space with an unconditional skipped-blocking decomposition, then
K contains such a bush provided K fails the point of continuity property.
This yields the earlier result, due to the authors (unpublished) and to W.
Schachermayer, that for closed bounded convex subsets of a Banach space with an
unconditional basis, the Krein-Milman property implies the point of continuity
property.
