A short note on hereditary Mazur intersection property

Definition 1. A Banach space (X, ∥ • ∥) is said to have the Mazur intersection property (MIP) if every closed, bounded, and convex set in (X, ∥ • ∥) is the intersection of the closed balls containing it.

MIP has been extensively studied over several decades and still continues to draw attention due to its rich geometric influence on the Banach space in question. A complete characterisation of MIP was obtained in [4,Theorem 2.1]. Earlier, Mazur himself [6] proved that Fréchet smoothness of (X, ∥ • ∥) implies MIP. The converse fails even in finite dimensions (see [1,Page 43]). But, a separable MIP space has separable dual [4, Theorem 2.1 (iii)], and hence, is an Asplund space. Therefore, it has a Fréchet smooth renorming [3,Theorem II.3.1]. Even this connection breaks down in non-separable Banach spaces. The fact that every Banach space can be isometrically embedded in a Banach space with MIP [8] adds some notoriety to the behaviour of spaces with MIP. In particular, this shows that MIP is not hereditary.

On the other hand, Fréchet smoothness is clearly a hereditary property. Thus, Mazur’s result actually shows that Fréchet smoothness of (X, ∥ • ∥) implies that it has the hereditary MIP, that is, X and all its subspaces have MIP. Now, in this form, is the converse of Mazur’s result true? That is, does the hereditary MIP imply Fréchet smoothness? Notice that if (X, ∥ • ∥) has the hereditary MIP, then (a) Every separable subspace of (X, ∥ • ∥) has MIP, and hence, has separable dual. Therefore, X is Asplund [10, Theorem 2.14]. (b) (X, ∥ • ∥) is smooth. To see this, note that for two dimensional spaces, smoothness and MIP are equivalent [9]. Also, if every two dimensional subspace of (X, ∥ • ∥) has MIP, and hence, is smooth, then (X, ∥ • ∥) is smooth [7,Propositon 5.4.21]. Hence, our question has an affirmative answer in finite dimensions. (c) The duality map on (X, ∥ • ∥) and all its subspaces are “quasicontinuous” [4, p 114]. Thus, it seems natural to conjecture that hereditary MIP implies Fréchet smoothness.

In this note, we obtain some sufficient conditions for hereditary MIP in terms of the set N (X) of non-Fréchet smooth points of S(X). In particular, we show that if (X, ∥ • ∥) is smooth and N (X) is finite, then (X, ∥ • ∥) has the hereditary MIP.

Borwein and Fabian in [2, Theorem 4] proved an interesting result that an infinite dimensional WCG Asplund space X admits an equivalent smooth norm ∥ • ∥ such that N (X) = {±x 0 } for some x 0 ∈ S(X, ∥ • ∥). Thus, our results show that the Borwein-Fabian norm gives a counterexample to the above conjecture.

A preliminary version of this note is contained in the second author’s Ph. D. thesis [5] written under the supervision of the first author. We would like to thank Prof. Gilles Godefroy for drawing our attention to [2].

Any undefined notions may be found in [3] or [7].

We consider real Banach spaces only. Let (X, ∥ • ∥) be a Banach space and X * its dual. By a subspace of (X, ∥•∥), we mean a closed linear subspace. For x ∈ X and r > 0, we denote by B(x, r) the open ball {y ∈ X : ∥x -y∥ < r} and by B[x, r] the closed ball {y ∈ X : ∥x -y∥ ≤ r} in (X, ∥ • ∥). We denote by B(X) the closed unit ball {x ∈ X : ∥x∥ ≤ 1} and by S(X) the unit sphere {x ∈ X : ∥x∥ = 1}. For x ∈ S(X), let

The multivalued map D X is called the duality map. Any selection of D X is called a support mapping.

We say that x ∈ S(X) is a smooth point if D X (x) is a singleton. A Banach space (X, ∥ • ∥) is smooth if every x ∈ S(X) is a smooth point.

We say that x ∈ S(X) is a Fréchet smooth point if the duality map D X is single-valued and norm-norm continuous at The following necessary or sufficient conditions for MIP follows from [4, Theorem 2.1] and are essentially already observed in [9].

Theorem 2. For a Banach space (X, ∥ • ∥), consider the following statements :

The set of extreme points of B(X * ) is norm dense in S(X * ).

A Banach space X is called Weakly Compactly Generated (WCG) if there is a weakly compact subset K of X such that span(K) = X.

In particular, all reflexive spaces and all separable spaces are WCG. Borwein and Fabian in [2, Theorem 4] proved that every infinite dimensional WCG Asplund space X admits an equivalent smooth norm ∥ • ∥ such that N (X) = {±x 0 } for some x 0 ∈ S(X, ∥ • ∥). In particular, every Banach space with a separable dual, e.g., c 0 , has an equivalent norm with the hereditary MIP, that is not Fréchet smooth.

Can N (X) be larger, retaining hereditary MIP? The following result gives a sufficient condition.

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