Geometry and Gâteaux smoothness in separable Banach spaces (Q2912193)

The paper is a survey on the interplay between differentiability of norms of real Banach spaces with the geometry of Banach spaces. Proofs of the stated theorems are sometimes omitted or sketched and many open problems are mentioned. We refer for the unexplained notions to the paper. The paper is divided into several sections.NEWLINENEWLINELet \((X,\|\cdot\|)\) be a Banach space. By a norm on \(X\), it is always meant an equivalent norm on \(X\).NEWLINENEWLINEThe first section studies pointwise directional Hölder derivatives. It is known that every separable Banach space can be (equivalently) renormed by a uniformly Gâteaux differentiable norm. The situation is different if it is required that the norm has directional Hölder derivative (pointwise or uniform). The following result summarizes some sufficient conditions for a space to be superreflexive (a space is superreflexive if it admits a uniformly Fréchet differentiable norm).NEWLINENEWLINELet \(X\) be a Banach space. Then each one of the following conditions implies that \(X\) is superreflexive. {\parindent=0.7cm\begin{itemize}\item[(i)] The space has the Radon-Nikodým property and admits a continuous bump function with pointwise directional Hölder derivative. \item[(ii)] Both \(X\) and \(X^*\) admit continuous bump functions with pointwise directional Hölder derivative. \item[(iii)] The space \(X\) admits a locally uniformly rotund norm with pointwise directional Hölder derivative on the sphere. \item[(iv)] The space \(X\) admits a bounded bump function with uniformly directional Hölder derivative. NEWLINENEWLINE\end{itemize}} Due to Pisier's result that every superreflexive Banach space has a norm with modulus of smoothness of power type [\textit{G. Pisier}, Isr. J. Math. 20, 326--350 (1975; Zbl 0344.46030)] and several other results, each of the conditions above characterizes superreflexive spaces.NEWLINENEWLINEThe second section of the paper focuses on the second-order Gâteaux differentiability. It is stated in the following theorem that twice differentiable Gâteaux norms are available in some separable superreflexive spaces.NEWLINENEWLINELet \(X\) be a separable Banach space with the Radon-Nikodým property. Then \(X\) admits a twice Gâteaux differentiable norm if and only if \(X\) admits a norm with modulus of smoothness of power type \(2\).NEWLINENEWLINEAs a corollary of a result from Section 1, one obtains the following assertion.NEWLINENEWLINELet \(X\) be a Banach space. Assume that both \(X\) and \(X^*\) admit continuous bump functions the restriction of which to each line in \(X\), respectively in \(X^*\), is twice differentiable. Then \(X\) is isomorphic to a Hilbert space.NEWLINENEWLINEThe third section studies preserved extreme points of nonempty bounded closed convex sets in Banach spaces and their relationship to the structure of spaces. Recall that preserved extreme points of a set \(C\subset X\) are elements of the set \(\mathrm{ext\,} \overline{C}^{w^*}\cap\mathrm{ext\,} C\) (here, \(\mathrm{ext\,}C\) denotes the set of extreme points of the set \(C\) and \(\overline{C}^{w^*}\) denotes the weak\(*\) closure of \(C\) in the double dual \(X^{**}\)).NEWLINENEWLINEFor a Banach space \(X\), the following conditions are equivalent. {\parindent=0.7cm\begin{itemize}\item[(i)] \(X\) does not have the Radon-Nikodým property. \item[(ii)] For every \(\varepsilon>0\) there exists a norm \(\|\cdot\|\) on \(X\) such that \(\text{dist}(\text{Ext}(B_{(X^{**},\|\cdot\|)}),X)\geq 1-\varepsilon\).NEWLINENEWLINESeveral theorems on the interplay between preserved extreme points of the unit ball \(B_X\) and the Radon-Nikodým property are stated. As an application, one gets the following assertion. NEWLINENEWLINE\end{itemize}} Let \(X\) be a Banach space such that, for every equivalent dual norm on \(X^*\), there is a point where the norm is Gâteaux differentiable with the derivative lying in \(X\). Then \(X\) has the Radon-Nikodým property.NEWLINENEWLINEThere is a connection between extreme points and the convex point of continuity property. \textit{W. Schachermayer}'s result [Trans. Am. Math. Soc. 303, 673--687 (1987; Zbl 0633.46023)] is mentioned:NEWLINENEWLINEAssume that a Banach space \(X\) has the Krein-Milman property and the convex point of continuity property. Then \(X\) has the Radon-Nikodým property.NEWLINENEWLINENext, a relation between reflexivity and preserved extreme points is investigated. It is shown, for example, that a Banach space \(X\) is not reflexive if and only if there exists a norm \(\|\cdot\|\) on \(X\) and an extreme point of \(B_{(X,\|\cdot\|)}\) that is not preserved.NEWLINENEWLINEThe fourth section deals with strongly Gâteaux differentiable norms. It is proved that every Banach space with a strongly Gâteaux differentiable norm is an Asplund space. Also, a separable Banach space \(X\) is reflexive if and only if every Gâteaux differentiable norm on \(X\) is strongly Gâteaux differentiable. Several other results concerning Gâteaux differentiablity and the convex point of continuity property are mentioned.NEWLINENEWLINEThe next section provides a survey on the interplay between variants of uniformities for rotund norms and poses several open problems. The last section gives a general geometric method of the construction of smooth norms whose dual norms are not strictly convex.