  We study some generalized metric properties of weak topologies when
restricted to the unit sphere of some equivalent norm on a Banach space, and
their relationships with other geometrical properties of norms. In case of dual
Banach space $X^*$, we prove that there exists a dual norm such that its unit
sphere is a Moore space for the weak$^*$-topology (has a G$_\delta$-diagonal
for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an
equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively).
