  In a recent paper a toy model (hypercubic model) undergoing a first-order
$\mathbb{Z}_2$-symmetry-breaking phase transition ($\mathbb{Z}_2$-SBPT) was
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 surfaces ($\Sigma_v$'s) of
configuration space. In this paper we show that at the origin of a
$\mathbb{Z}_2$-SBPT there is a geometric property of the $\Sigma_v$'s, i.e.,
dumbbell-shaped $\Sigma_v$'s suitably defined, which includes a topological
change as a limiting case. This property is necessary and sufficient condition
to entail a $\mathbb{Z}_2$-SBPT. This new approach has been applied to three
models: a modified version introduced here of the hypercubic model, a model
introduced in a recent paper with a continuous $\mathbb{Z}_2$-SBPT belonging to
several universality classes, and finally to a physical models, i.e., the
mean-field $\phi^4$ model and a simplified version of it.
