Generalized metric properties of spheres and renorming of Banach spaces

A remark on totally smooth renormings

“…Banach spaces that satisfy one (and then all) of the conditions (i) to (iv) in Theorem 1 have been characterized in other different ways. Let us mention here that, for example, Theorem 1.4 in [3] provides a few of them, in terms of (a) the existence of a dual norm in X * such that (S X * , w * ) is a Moore space, or (b) the existence of an equivalent dual norm such that (S X * , w * ) is symmetrized by a symmetric ρ such that every point x * ∈ S X * has w * -neighborhoods of arbitrary small ρ-diameter, or (c) the existence of a dual equivalent norm such that (S X * , w * ) is metrizable, or even that (d) that (B X * , w * ) is a descriptive compact space (for details, see the op. cit.…”

A topological characterization of dual strict convexity in Asplund spaces

“…These properties have been studied in a purely topological context, as well as in relation to strictly convex norms on Banach spaces. We refer the reader to [2,3,4,6,8,9] and the references therein for more details.…”

A note on weak-star and norm Borel sets in the dual of the space of continuous functions

“…Many authors have studied the relationship between the Borel sets generated by the weak topology and the one generated by the norm topology (see for example [2,3,14,17]) and have found various conditions for the coincidence of this two classes. In particular it is shown that the coincidence of the above σ-algebras is related to the existence of special types of equivalent norms (see [7,8,9,13,14,17], for a study of these types of equivalent renormings).…”