## Abstract

We show that for a separable Banach space X failing the Radon-Nikodým property (RNP), and ε > 0, there is a symmetric closed convex subset C of the unit ball of X such that every extreme point of the weak-star closure of C in the bidual X** has distance from X bigger than 1 -ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, for ε > 0, a convex set C of diameter equal to 1 such that each slice of C has diameter bigger than 1 -ε. Some more related results about the geometry of Banach spaces failing RNP are given.

Original language | English (US) |
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Pages (from-to) | 225-257 |

Number of pages | 33 |

Journal | Israel Journal of Mathematics |

Volume | 65 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 1989 |