Generalized metric properties of spheres and renorming of Banach spaces

We study the equivalence under renorming of several geometric and topological properties of the unit sphere of a Banach space with respect to weak topologies. The geometric properties considered are stronger forms of rotundity and the topological properties are generalizations of metrizability. In the case of dual Banach spaces endowed with the weak\(^*\) topology our results provide a full understanding of the rotund case and complement our previous work on \(w^*\)-LUR renorming.