A characterization of Banach spaces containing $\ell_1(\kappa)$ via ball-covering properties

TL;DR

This paper extends Godefroy's characterization of Banach spaces containing  to higher cardinalities, linking ball-covering properties with the existence of (\u03ba^+) subspaces and octahedral norms.

Contribution

It generalizes Godefroy's result to higher cardinalities, establishing a connection between ball-covering properties and (\u03ba^+) subspaces in Banach spaces.

Findings

Banach spaces contain (ba^+) iff they can be renormed so their unit sphere cannot be covered by ba many balls.

Ball-covering properties characterize the presence of (ba^+) subspaces.

Relation between ball-coverings and octahedral norms is extended to higher cardinalities.

Abstract

In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of $\ell_1$ if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy's result to higher cardinalities. We prove that, for an infinite cardinal $\kappa$, a Banach space $X$ contains an isomorphic copy of $\ell_1(\kappa^+)$ if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by $\kappa$ many open balls not containing $\alpha B_X$, where $\alpha\in (0,1)$. We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.