Non-smooth Analysis, Optimisation Theory and Banach Space Theory

The questions listed here do not necessarily represent the most significant problems from the areas of Non-smooth Analysis, Optimisation theory and Banach space theory, but rather, they represent a selection of problems that are of interest to the authors. 1. Weak Asplund spaces Let X be a Banach space. We say that a function φ : X → R is Gâteaux differentiable at x ∈ X if there exists a continuous linear functional x∗ ∈ X∗ such that x∗(y) = lim λ→0 φ(x+ λy)− φ(x) λ for all y ∈ X. In this case, the linear functional x∗ is called the Gâteaux derivative of φ at x ∈ X. If the limit above is approached uniformly with respect to all y ∈ BX -the closed unit ball in X, then φ is said to be Fréchet differentiable at x ∈ X and x∗ is called the Fréchet derivative of φ at x. A Banach space X is called a weak Asplund space [Gâteaux differentiability space] if each continuous convex function defined on it is Gâteaux differentiable at the points of a residual subset (i.e., a subset that contains the intersection of countably many dense open subsets of X) [dense subset] of its domain. Since 1933, when S. Mazur [55] showed that every separable Banach space is weak Asplund, there has been continued interest in the study of weak Asplund spaces. For an introduction to this area see, [61] and [32]. Also see the seminal paper [1] by E. Asplund. The main problem in this area is given next. Question 1.1. Provide a geometrical characterisation for the class of weak As1001 ? plund spaces. Note that there is a geometrical dual characterisation for the class of Gâteaux differentiability spaces, see [67, §6]. However, it has recently been shown that there are Gâteaux differentiability spaces that are not weak Asplund [58]. Hence The first named author was supported by NSERC and by the Canada Research Chair Program. The second named author was supported by a Marsden Fund research grant, UOA0422, administered by the Royal Society of New Zealand.