On the structure of non-dentable closed bounded convex sets
Let X be a Banach space and W a closed convex subset of X. W is said to have the CPCP (convex point of continuity property) [KMP (Krein-Milman property)] if every nonempty closed bounded convex subset K of W has a weak-norm point of continuity relative to K [ext \(K\neq \emptyset]\). The main results of the paper are as follows. 1) if X fails the RNP (Radon-Nikodým property) but satisfies the CPCP, then there exists a subspace Y of X so that Y has an FDD (finite- dimensional decomposition) but fails the KMP. 2) if K is a nonempty closed bounded convex subset of X so that K is non- dentable (i.e. \(k\in cocl(K\setminus B(k,\epsilon))\) for some \(\epsilon >0\) and \(k\in K)\) and has SCSP (small combination of slices property; i.e. K contains a convex combination of slices of K which is of arbitrarily small diameter), then there is a nonempty closed convex subset W of K such that ({\S} ) W is non-dentable and the weak and norm topologies on W coincide. (Moreover, there exists a subspace Y of X so that Y has an FDD and a nonempty bounded closed convex subset W satisfying ({\S} ).) These (and other related) results extend the results and ideas of J. Bourgain, N. Ghoussoub, G. Godefroy, B. Maurey and others. The exposition is self- contained.
scientific article; zbMATH DE number 4067603
scientific article