  Recently a new Bell inequality has been introduced (CGLMP,KKCZO) that is
strongly resistant to noise for maximally entangled states of two
$d$-dimensional quantum systems. We prove that a larger violation, or
equivalently a stronger resistance to noise, is found for a non-maximally
entangled state. It is shown that the resistance to noise is not a good measure
of non-locality and we introduce some other possible measures. The
non-maximally entangled state turns out to be more robust also for these
alternative measures. From these results it follows that two Von Neumann
measurements per party may be not optimal for detecting non-locality. For
$d=3,4$, we point out some connections between this inequality and
distillability. Indeed, we demonstrate that any state violating it, with the
optimal Von Neumann settings, is distillable.
