Extremal Property of the Square Lattice

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.