227 research outputs found
The equivalence between CPCP and strong regularity under Krein-Milman property
We obtain a result in the spirit of the well-known W. Schachermeyer and H. P.
Rosenthal research about the equivalence between Radon-Nikodym and Krein-Milman
properties, by showing that, for closed, bounded and convex subsets C of a
separable Banach space, under Krein-Milman property for , one has the
equivalence between convex point of continuity property and strong regularity
both defined for every locally convex topology on C, containing the weak
topology on C. Then, under Krein-Milman property, not only the classical convex
point of continuity property and strong regularity are equivalent, but also
when they are defined for an arbitrary locally convex topology containing the
weak topology. We also show that while the unit ball of fails convex
point of continuity property and strong regularity (both defined for the weak
topology), threre is a locally convex topology on , containing the
weak topology on , such that still fails convex point of continuity
property for , but surprisingly enjoys strong regularity for
-open sets. Moreover, satisfies the diameter two property for the
topology , that is, every nonempty -open subset of has diameter
two even though every -open subset of contains convex combinations of
relative -open subsets with arbitrarily small diameter, that is,
fails the strong diameter two property for the topology . This stresses
the known extreme differences up to now between those diameter two properties
from a topological point of view.Comment: 19 page
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