  In a recent paper a toy model (called hypercubic model) undergoing a
first-order $\mathbb{Z}_2$ symmetry breaking phase transition (SBPT) has been
introduced. The hypercubic model was inspired by the \emph{topological
hypothesis}, according to which a phase transition may be entailed by suitable
topological changes of the equipotential hypersurfaces $\Sigma_v$ of
configuration space. The $\Sigma_v$'s of the hypercubic model have a single
topological change, which, under further particular hypotheses of geometric
nature, entails the $\mathbb{Z}_2$-SBPT. In this paper we introduce an extended
version of the hypercubic model in which no topological change in the
$\Sigma_v$'s is present anymore, but nevertheless the $\mathbb{Z}_2$-SBPT
occurs the same. We introduce a geometric property of the $\Sigma_v$'s (i.e.
dumbbell $\Sigma_v$'s suitably defined) that is sufficient to entail a
$\mathbb{Z}_2$-SBPT regardless their topology. The paper ends by applying the
picture of the dumbbell $\Sigma_v$'s to a physical model, i.e. the mean-field
$\phi^4$ model.
