  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.

Functional Analysis