  A result of G. Godefroy asserts that a Banach space $X$ contains an
isomorphic copy of $\ell_1$ if and only if there is an equivalent norm
$|||\cdot|||$ such that, for every finite-dimensional subspace $Y$ of $X$ and
every $\varepsilon>0$, there exists $x\in S_X$ so that $|||y+r x|||\geq
(1-\varepsilon)(|||y|||+\vert r\vert)$ for every $y\in Y$ and every
$r\in\mathbb R$. In this paper we generalize this result to larger cardinals,
showing that if $\kappa$ is an uncountable cardinal then a Banach space $X$
contains a copy of $\ell_1(\kappa)$ if and only if there is an equivalent norm
$|||\cdot|||$ on $X$ such that for every subspace $Y$ of $X$ with
$dens(Y)<\kappa$ there exists a norm-one vector $x$ so that $||| y+r
x|||=|||y|||+\vert r\vert$ whenever $y\in Y$ and $r\in\mathbb{R}$. This result
answers a question posed by S. Ciaci, J. Langemets and A. Lissitsin, where the
authors wonder whether the previous statement holds for infinite succesor
cardinals. We also show that, in the countable case, the result of Godefroy
cannot be improved to take $\varepsilon=0$.
