On the structure of non-dentable closed bounded convex sets

Abstract

A self-contained proof is given of the following result. Theorem. Let K be a non-dentable closed bounded convex nonempty subset of a Banach space X so that K equals the closed convex hull of its weak-to-norm points of continuity. There exists a nonempty subset W of K satisfying W is non-dentable closed convex and the weak and norm topologies on W coincide. (∗) Moreover there exists a closed linear subspace Y of X so that Y has a Finite-Dimensional Decomposition and a nonempty convex subset W satisfying (∗). (Any bounded set W satisfying (∗) has no extreme points.) This yields the recent discovery of W. Schachermayer that a closed bounded convex subset of a Banach space has the Radon-Nikodým Property (RNP) provided it has the Convex Point of Continity Property (CPCP) and the Krein-Milman Property (KMP), and the CPCP case of the earlier discovery of J. Bourgain that a Banach space has the RNP provided every subspace with a Finite-Dimensional Decomposition has the RNP. The proof is a variation and crystallization of the original arguments for these discoveries. In particular, the proof uses the concept of convex sets with small combinations of slices. This is a refinement of the concept of strong regularity, introduced by N. Ghoussoub, G. Godefroy, and B. Maurey; both concepts crystallize ideas appearing in Bourgain's work. The proof also yields further results concerning the structure of non-dentable convex sets.