  Motivated by a 2019 result of Faulhuber-Steinerberger, we demonstrate that
the real square lattice $\mathbb{Z}^2$ exhibits the same local, extremal
property as the hexagonal lattice $\Lambda$, where distances of lattice points
from the `deep holes' of natural fundamental domains increase under
perturbation. If $\Delta$ is a small perturbation of $\mathbb{Z}^2$ in the
space of unimodular lattices, consider $C_r$, the set of points in $A_r$
shifted to $\Delta$. If $\Delta$ is a perturbation of the lattice
$\mathbb{Z}^2$ with respect to the Euclidean metric, then for a fixed deep hole
$p$, the summed total distance of lattice points to $p$ strictly increases, and
is bounded below by a function of the distance between the lattice and its
perturbation. Additionally, we show this growth is approximately preserved by
convex functions.
