Characterisation of normed linear spaces with Mazur's intersection property

Abstract

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak * denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties. It was Mazur [7] who drew attention to the euclidean space property (I): every bounded closed convex set can be represented as an intersection of closed balls; and he began the investigation to determine those normed linear spaces which possess this property. Phelps [9] continued this investigation, characterising finite dimensional spaces with property (I). Recently, Sullivan [/2] has given a characterisation of smooth spaces with property (I).

A note on norm attaining functionals

“…A Banach space is said to have the Mazur Intersection Property [18] if every bounded closed convex set is an intersection of closed balls. Analogously, a dual Banach space has the Weak* Mazur Intersection Property [8] if every weak* compact convex set is an intersection of closed dual balls. A Banach space (X, · ) has the Mazur Intersection Property if, and only if, the set of weak* denting points of B · * is dense in S · * .…”

Mazur's intersection property of balls for compact convex sets

“…This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.It was proved by Phelps in [5 3 that for a finite dimensional Banach space X the set of all extreme points of the dual unit ball B* is dense in the unit sphere S* <= X* if and only if X has the following property called here property (CD : every compact convex set G in X is an intersection of closed balls.We extend the necessity part of this result to general Banach spaces (Theorem 1), by using significantly ideas of Giles, Gregory and Sims in [3].We then prove that every Banach space with a transfinite Schauder …”

“…The C-topology on X* will mean the topology of uniform convergence on compact sets in X . For a set A c X* , the closure of A in the C-topology will be denoted by We will use the following "compact" version of a Definition in [3], DEFINITION 1. If G is a compact convex symmetric set in a Banach space X and e > 0 t we say that a point x e S~ c X belongs to the set M^ if there is a 6 > 0 such that …”

“…An adjustment of t h a t of Lemma 2.2 in [3] ((i) => i i ) ) . We are t o show t h a t i f for some f e S* and some compact convex s e t G c X where ll/ll is the original dual norm on X* and || • || is Day's norm on «, (T) (see for example [63,[7D).…”

“…We extend the necessity part of this result to general Banach spaces (Theorem 1), by using significantly ideas of Giles, Gregory and Sims in [3].…”