A variational principle for Gaussian lattice sums

We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices Λ ⊂ R² with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be interpreted as the dual of a 1988 result of Montgomery who proved that the hexagonal lattice minimizes the maximal values. Our inequality resolves a conjecture of Strohmer and Beaver about the operator norm of a certain type of frame in L²(R). It has implications for minimal energies of ionic crystals studied by Born, the geometry of completely monotone functions and a connection to the elusive Landau constant.