  Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups
of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the
subgroups in $\mathcal{M}$ is not equal to the intersection of any proper
subset. The minimal dimension of $G$, denoted ${\rm Mindim}(G)$, is the minimal
size of a maximal irredundant set of maximal subgroups of $G$. This invariant
was recently introduced by Garonzi and Lucchini and they computed the minimal
dimension of the alternating groups. In this paper, we prove that ${\rm
Mindim}(G) \leqslant 3$ for all finite simple groups, which is best possible,
and we compute the exact value for all non-classical simple groups. We also
introduce and study two closely related invariants denoted by $\alpha(G)$ and
$\beta(G)$. Here $\alpha(G)$ (respectively $\beta(G)$) is the minimal size of a
set of maximal subgroups (respectively, conjugate maximal subgroups) of $G$
whose intersection coincides with the Frattini subgroup of $G.$ Evidently,
${\rm Mindim}(G) \leqslant \alpha(G) \leqslant \beta(G)$. For a simple group
$G$ we show that $\beta(G) \leqslant 4$ and $\beta(G) - \alpha(G) \leqslant 1$,
and both upper bounds are best possible.
