  We compute the global dimension function $\mathrm{gldim}$ on the principal
component $\mathrm{Stab}^{\dag}(\mathbb{P}^2)$ of the space of Bridgeland
stability conditions on $\mathbb{P}^2$. It admits $2$ as the minimum value and
the preimage $\mathrm{gldim}^{-1}(2)$ is contained in the closure
$\bar{\mathrm{Stab}^{\mathrm{Geo}}(\mathbb{P}^2)}$ of the subspace consisting
of geometric stability conditions. We show that $\mathrm{gldim}^{-1}[2,x)$
contracts to $\mathrm{gldim}^{-1}(2)$ for any real number $x\geq 2$ and that
$\mathrm{gldim}^{-1}(2)$ is contractible.
