  We introduce a new topological property called (*) and the corresponding
class of topological spaces, which includes spaces with $G_\delta$-diagonals
and Gruenhage spaces. Using (*), we characterise those Banach spaces which
admit equivalent strictly convex norms, and give an internal topological
characterisation of those scattered compact spaces $K$, for which the dual
Banach space $C(K)^*$ admits an equivalent strictly convex dual norm. We
establish some relationships between (*) and other topological concepts, and
the position of several well-known examples in this context. For instance, we
show that $C(\mathcal{K})^*$ admits an equivalent strictly convex dual norm,
where $\mathcal{K}$ is Kunen's compact space. Also, under the continuum
hypothesis CH, we give an example of a compact scattered non-Gruenhage space
having (*).

Functional Analysis;General Topology