## Doctoral Theses

### The Mazur Intersection Property in Banach Spaces and Related Topics.

2-22-1991

2-22-1992

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Roy, Ashoke Kumar (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In the first part of this chapter we explain in general terms the main theme of this thesis and provide a chapter wise summary of its principal results. The second part recapitulates some of the known notions and results used in the subsequent chapters. The numbers given in parentheses correspond to those in the list of references on page 70.S. Mazur (40] was the first to consider the following smoothness property in normed linear spaces, called the Mazur Intersection Property (MIP), or, more briefly, the Property (1):Every closed bounded convex set is the intersection of closed balls containing it.He showed that any reflexive Banach space with a FrÃ©chet-differentiable norm has this property.Later, R. R. Phelps (42] provided a more geometric insight into this property by showing that(a) A normed linear space X has the MIP if the w*-strongly exposed points of the unit ball B(X*) of the dual X are norm dense in the unitsphere S. (X*)(b) If a normed linear space X has the MIP, every support mapping on X maps norm dense subsets of S(X) to norm dense subsets of S(X*).(c) A finite dimensional normed linear space X has the MIP if and only if the extreme points of B(X*) are norm dense in s(X*).He also asked whether the sufficient condition (a) is also necessary. To date, this remains an open question.Nearly two decades later, Phelps' characterisation (c) was extended by J. R. Giles, D. A. Gregory and B. Sims (21] to general normed linear spaces, developing an idea due to F. Sullivan (51), and they proved, inter alia,Theorem 1.1 For a normed linear space X, the following are equivalent :(a) The w*-denting points of B(X*) are norm dense in S(X*).(b) X has the MIP. (c) Every support mapping on X maps norm dense subsets of S(X) to norm dense subsets of S(X*).They also showed that in dual Banach spaces, the MIP implies reflexiv- ity and considered the weaker property that Ã©very weak* compact convex set in a dual space is the intersection of balls (w*-MIP). Investigating the necessity of Phelps' condition (a), they showed that it is indeed necessary if, in addition, the dual X has the w*-MIP, or, X is an Asplund space. They now asked whether the MIP necessarily implies Asplund. To date, this also remains open.Notice that if X is separable and has the MIP, Phelps' condition (c) (or, Theorem 1.1(c) above) implies that it has a separable dual and hence is Asplund. So one asks, is the MIP hereditary, i.e., inherited by subspaces ? The answer, unfortunately, is no. In Chapter 4, we give an example and discuss the subspace question in more detail.However, since the Asplund Property is invariant under equivalent renorming, a more pertinent question is whether the existence of an equivalent norm with the MIP is hereditary. Some discussions on MIP-related renorming questions may be found in [9], [47), [54] and [57). However, in this work, we do not discuss any renorming problem but concentrate instead on some of the isometric questions that arise.

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