An Extremal Property of the Hexagonal Lattice

Abstract

We describe an extremal property of the hexagonal lattice $\Lambda \subset {\mathbb{R}}^2$. Let $p$ denote the circumcenter of its fundamental triangle (a so-called deep hole) and let $A_r$ denote the set of lattice points that are at distance $r$ from $p$ \begin{equation} A_r = \left\{ λ\in Λ: \| λ- p \| = r\right\}. \end{equation} If $\Gamma$ is a small perturbation of $\Lambda$ in the space of lattices with fixed density and $C_r$ denotes the set of points in $A_r$ shifted to the new lattice, then \begin{equation} \sum_{μ\in C_r}{ \| p - μ\|} - \sum_{λ\in A_r}{ \| p - λ\|} \gtrsim r \, |A_r| \, d(Λ, Γ)^2, \end{equation} where $d(\Lambda ,\Gamma )$ denotes the distance between the lattices: the hexagonal lattice has the property that `far away points are closer than they are for nearby lattices'. This has implications in the calculus of variations: assume \begin{equation} g_Γ(z) = \sum_{γ\in Γ} f( \|z - γ\|) \quad \mbox{ satisfies } \quad \min_{z \in \mathbb{R}^2} g_Λ(z) = g_Λ(p). \end{equation} For a certain class of compactly supported functions $f$, the hexagonal lattice $\Lambda$ is then a strict local maximizer of \begin{equation} \max_Γ \min_{z \in \mathbb{R}^2} \sum_{γ\in Γ}{f( \|z - γ\| )}, \end{equation} where the maximum runs over all lattices of fixed density.