Generalized metric properties of spheres and renorming of normed spaces

TL;DR

This paper explores generalized metric properties of spheres in Banach spaces under equivalent norms, linking these properties to the existence of specific dual norms with desirable topological and geometric features.

Contribution

It establishes a characterization of dual Banach spaces admitting weak*-LUR or rotund dual norms via the generalized metric properties of their unit spheres.

Findings

Existence of dual norms with Moore space property for the weak* topology.

Characterization of dual spaces with weak*-LUR dual norms.

Connection between geometric properties of norms and topological properties of spheres.

Abstract

We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove that there exists a dual norm such that its unit sphere is a Moore space for the weak$^*$-topology (has a G$_\delta$-diagonal for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively).