Generalized metric properties of spheres and renorming of Banach spaces (Q2317596)

Let \(X\) be a Banach space and denote by \(S_X\) its unit sphere. The authors study the equivalence under renorming of stronger forms of rotundity and generalizations of metrizability. A particular example of the type of theorems contained in this paper is the following: The dual \(X^\ast\) of a Banach space \(X\) admits an equivalent dual weak*-LUR norm if and only if \(X^\ast\) admits an equivalent dual norm \(\|\cdot\|\) such that there are weak* open coverings \(\mathcal{G}_1, \mathcal{G}_2,\dots\) of \(S_{X^\ast}\) such that, for every \(x^\ast\) in \(S_{X^\ast}\), the sets \(\bigcup\{U\in \mathcal{G}_n, \,x^\ast\in U\}\), \(n\in \mathbb{N}\), form a neighborhood basis at \(x^\ast\).