An Extremal Property of the Hexagonal Lattice

Abstract

We describe an extremal property of the hexagonal lattice Λ ⊂ R 2 . Let p denote the circumcenter of its fundamental triangle (a so-called deep hole) and let Ar denote the set of lattice points that are at distance r from p Ar = {λ ∈ Λ : λ − p = r} .If Γ is a small perturbation of Λ in the space of lattices with fixed density and Cr denotes the set of points in Ar shifted to the new lattice, thenwhere d(Λ, Γ) 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: assumeFor a certain class of compactly supported functions f , the hexagonal lattice Λ is then a strict local maximizer ofwhere the maximum runs over all lattices of fixed density.2010 Mathematics Subject Classification. 52C05, 74G65.

Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies

“…2.5]). Furthermore, we remark that a similar result to Corollary 1.2, involving local maximality of the hexagonal lattice among lattices, has been established in [32]. It might well be true that the result in [32] holds for completely monotone functions, as the technical assumptions there look as if they can be tailored to suit completely monotone functions.…”

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“…An important conjecture by Strohmer and Beaver [58] expects the condition number of (deterministic) Gabor frames with Gaussian windows to be optimized by a hexagonal lattice. Significant progress towards a proof has been made by Faulhuber and collaborators [26], and a preprint with a full solution of the problem has recently been posted in [18].…”

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“…The best result available at the moment is the result by Montgomery [61], which, among other results, implies the universal optimality of the hexagonal lattice among lattices. Also, it has to be noticed that the class of potentials for which the hexagonal lattice is optimal at all densities is expected to be bigger than the one of completely monotone functions, see [10] and compare also [38]. Furthermore, for a large class of potentials the results in [61] imply that the hexagonal lattice is a so-called ground state among all possible lattices for the problem of energy minimization of pairwise interacting particles.…”