Perfect, strongly eutactic lattices are periodic extreme

We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension $d\leq 8$ and $d=24$.

💡 Research Summary

The paper introduces a systematic framework for studying periodic sphere packings that are formed by the union of a finite number of translates of a lattice. The authors define a parameter space 𝒫_{d,m} consisting of all point sets P = ⋃_{i=1}^{m}(L + v_i) where L ⊂ ℝ^d is a full‑rank lattice, v_1 = 0 and the remaining v_i are translation vectors. Within this space the packing density 𝛿(P) is expressed as 𝛿(P) = κ_d r_min(P)^d / vol(L), where κ_d is the volume of the unit d‑ball and r_min(P) denotes the minimal inter‑point distance. By treating the lattice basis matrix B and the translation matrix V as continuous variables, the authors develop first‑ and second‑order variations of the density function.

A central technical contribution is the derivation of explicit formulas for the first and second variations of 𝛿. The first variation vanishes precisely when the underlying lattice L satisfies two classical conditions from the theory of extreme lattices: perfection and strong eutaxy. Perfection means that the set of minimal vectors M(L) spans the full space of symmetric d×d matrices via their outer products; strong eutaxy means that there exist positive weights c_x (x ∈ M(L)) such that Σ_{x∈M(L)} c_x x x^T = I_d. Under these hypotheses the linear term in the variation of 𝛿 disappears for any infinitesimal change of B and V.

The second variation splits into a lattice part and a translation part. The lattice part is a quadratic form built from the Gram matrix of the minimal vectors weighted by the eutaxy coefficients; because of strong eutaxy this form is positive semidefinite. The translation part encodes how the distances between distinct translates change under perturbations; the authors show that the constraints imposed by keeping r_min(P) unchanged force this part to be non‑positive as well. Consequently, when L is perfect and strongly eutactic, the Hessian of 𝛿 at any point (B,V) in 𝒫_{d,m} is negative semidefinite, i.e. the density cannot increase under any infinitesimal periodic deformation.

The main theorem, therefore, states: If a lattice L is perfect and strongly eutactic, then no periodic point set obtained by adding any finite number of translates of L can have a strictly larger packing density. In the language introduced by the authors, such lattices are “periodic extreme.” The proof proceeds by showing that the first variation vanishes (perfection) and the second variation is non‑positive (strong eutaxy together with the translation constraints) for all m ≥ 1.

The authors apply this result to the known densest lattice packings in low dimensions. In dimensions d ≤ 8 the lattices A_d, D_d, E_8 and their relatives are perfect and strongly eutactic; in dimension 24 the Leech lattice enjoys the same properties. Hence these packings are not only locally optimal among lattices but also among all periodic packings, confirming that no improvement can be achieved by introducing a finite number of additional translates.

Beyond the immediate applications, the paper opens several avenues for further research. While perfection and strong eutaxy are sufficient for periodic extremality, it remains unknown whether they are necessary. The behavior of non‑perfect or merely eutactic lattices under periodic perturbations is an open problem. Moreover, the authors suggest developing computational tools to explore the global landscape of 𝒫_{d,m} and to compare periodic extreme lattices with genuinely non‑lattice periodic packings (e.g., quasicrystalline or aperiodic structures). Extending the variational framework to incorporate additional constraints such as symmetry groups or to handle infinite families of translates could also deepen our understanding of high‑dimensional sphere packing.

In summary, the paper provides a rigorous variational proof that perfect, strongly eutactic lattices are immune to any density‑increasing periodic deformation. This result unifies and extends classical extremality theory, confirming that the celebrated optimal lattices up to dimension eight and the Leech lattice in dimension twenty‑four are not only lattice‑optimal but also optimal among all periodic packings.