On the structure of non-dentable closed bounded convex sets
type: Item
instance of: scholarly article
title: On the structure of non-dentable closed bounded convex sets
zbMATH Open document ID: 0654.46024
DOI: 10.1016/0001-8708(88)90050-3
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 4067603
zbMATH Keywords: dentable set
zbMATH Keywords: finite-dimensional decomposition
zbMATH Keywords: small combinations of slices
zbMATH Keywords: closed convex subset
zbMATH Keywords: convex point of continuity property
zbMATH Keywords: Krein- Milman property
zbMATH Keywords: Radon-Nikodým property
author: Haskell P. Rosenthal
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90050-3
OpenAlex ID: W2045973506
cites work: Dentability and finite-dimensional decompositions
cites work: Q3857296
cites work: Sets with the Radon-Nikodým property in conjugate Banach space
cites work: Q3913880
cites work: Geometric aspects of convex sets with the Radon-Nikodym property
cites work: Q4148368
cites work: Some topological and geometrical structures in Banach spaces
cites work: A Counterexample to a Problem on Points of Continuity in Banach Spaces
cites work: A counterexample to several questions about Banach spaces
cites work: Martingale proofs of some geometrical results in Banach space theory
cites work: Q3976524
cites work: Weak *-Polish Banach spaces
cites work: Q3976518
cites work: The Krein-Milman property and a martingale coordinatization of certain non-dentable convex sets
cites work: The Radon-Nikodym Property and the Krein-Milman Property are Equivalent for Strongly Regular Sets
cites work: Q3025646
description: scientific article; zbMATH DE number 4067603