Paper detail

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

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 $κ$, a Banach space $X$ contains an isomorphic copy of $\ell_1(κ^+)$ if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by $κ$ many open balls not containing $αB_X$, where $α\in (0,1)$. We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.