In this talk we are going to study Gaussian functions summed on 2-dimensional, shifted lattices. A classical result of H. Montgomery from 1988 states that, for all fixed densities, the hexagonal lattice minimizes among all lattices the maximal value with respect to the shifts. Recently, in a collaboration with L. Bétermin and S. Steinerberger, we were able to prove the analogous result, namely that the hexagonal lattice also maximizes among all lattices the minimal value among all shifts of the lattice. However, the problem of maximizing the minimum is much more subtle. While the maximum of a Gaussian lattice sum is always attained at lattice points, there is little control over the shift where the minimal value is attained. In general, the minimal value also wanders as the density changes. Just like Montgomery's result, our result has a lot of implications: our inequality resolves the conjecture of Strohmer and Beaver about the operator norm of a Gaussian frame operator, 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.
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