Universally Optimal Periodic Configurations in the Plane

Universally optimal periodic configurations in the plane, Discrete Analysis 2025:22, 63 pp.

Let $\omega_n$ be a configuration of $n$ points in $\mathbb R^d$. We define the energy of $\omega_n$ to be
$$
E_F(\omega_n)=\sum_{j\neq i} F(x_i-x_j),
$$
where $F:\mathbb R^d\to \mathbb R$ is a potential function. We are interested in the case where $F$ is periodic with respect to some $d$-dimensional lattice $\Lambda$. For the purpose of this paper, we specialize to the case when
$$
F_a(x)=\sum_{v\in\Lambda} e^{-a|x+v|^2},\ a>0,
$$
is a $\Lambda$-periodic sum of Gaussians.

We say that a configuration $\omega_n$ is optimal with respect to $F_a$ as above if it minimizes the emergy $E_{F_a}$ over all possible $n$-point configurations in $\mathbb R^d$. We further say that $\omega_n$ is $\Lambda$-_universally optimal_ if it is optimal for all $a>0$.

Passing from a property of point configurations to a property of the lattice itself, we say that $\Lambda$ is _universally optimal_ if for any sublattice $\Phi$ of $\Lambda$, the configuration $\omega(\Phi,\Lambda):=\Lambda/\Phi$ (for any choice of representatives) is $\Lambda$-universally optimal.

Currently, only three examples of universally optimal lattices are known: the lattice $\mathbb Z$ in $\mathbb R$ (due to Cohn and Kumar), and the $E_8$ and Leech lattices in, respectively, $\mathbb R^8$ and $\mathbb R^{24}$ (due to Cohn, Kumar, Miller, Radchenko, and Viazovska). Cohn and Kumar conjectured that the hexagonal lattice in 2 dimensions should also be universally optimal. The hexagonal lattice enjoys many properties that make it a natural candidate: for example, it has long been known to be the optimal configuration for unit circle packing in the plane. However, the problem remains open.

The present paper proves a partial result in the direction of the conjecture, demonstrating that certain natural configurations are universally optimal with respect to the hexagonal lattice. This would follow immediately if the above conjecture were to be confirmed; conversely, proving universal optimality for a larger family of configurations of the same type would prove the conjecture.

The proof is based on linear programming: the main idea is that the energy of a point configuration may be bounded from below in terms of the Fourier coefficients of an appropriately chosen auxiliary function (sometimes called a _witness function_). The main challenge is to construct the optimal auxiliary functions, and that is what this paper accomplishes.